I've only read Ender's Game and Ender's Shadow so please use spoiler tags if necessary. :) I won't use them for my initial post though, since then everything would be in spoiler tags.
Ender: LSI
Often typed as ILI just because he's intelligent, Ender seems likely to actually be LSI, especially when contrasted with Bean. In Battle School and Command School, Ender magnificently adapts to the situations thrown at him where he is vastly disadvantaged. He does so on the spot, within seconds/minutes of him seeing the situation, demonstrating that he is excellent at tactics. In contrast, he actually shows little consideration for long-term strategy, not thinking about an overall plan for defeating the Formics or even trying to figure out exactly why the teachers are training him the way that they are. He is put in a system, and he excels in the system without trying to understand the big picture point of it. His sheer intelligence and Ni HA are enough to come up with "the enemy's gate is down" but not enough to manipulate the general flow to what he wants. He forcefully, successfully rows down a challenging stream, but the teachers (Third Invasion), Valentine (preventing Ender from ending up in Peter's grasp), and Bean (picking Ender's army, watching over him closely) mostly decide the direction of the stream. When confronted by Stilson and Bonzo, he skillfully physically overcomes them despite being smaller. Note his victory over Bonzo especially indicates Se ego over Ni ego. Ender did not try to manipulate the confrontation to occur at a favorable time for him--he was absentmindedly caught in the shower. However, presented with immediate danger, he reacts quickly to over come it. Contrast this with how Bean deals with his challenge from the teachers: Bean plants an intricate trap for Achilles, ensuring that the confrontation puts Bean in a favorable position, knowing that he would lose if confronted by Achilles on his own terms.
Bean: LII
Bean reacts to the Battle School system in a completely different way from Ender. Bean sees the system--the world, even--as a puzzle to be understood/solved; then he will decide for himself how much he wants to cooperate. He loathes the idea of being forced to obey stupid people, not because he hates not having power (he happily leaves Ender with all the official power, helping him quietly, without fanfare or demanding recognition), but because he hates being forced to do stupid things. His sheer intelligence and practice allow him to be a great soldier in Ender's army, but both of them agree he is not the best soldier on the field--only his mind is the fastest. He beats Achilles with strategy, but even then does not have the volitional force to kill him, only to have him committed to a mental institution. He tries to prevent Bonzo's gang from attacking Ender with a rousing speech rather than force. He understands the situation on Earth by reading as much about political theory and current events as he can and then perceives the situation better than anyone else, allowing him to make a suggestion to Locke and Demosthenes that they both actually take. This all points to INTx.
For LII over ILI, consider that he ultimately figures out that they are the Third Invasion, now, by noticing a seemingly small inconsistency--if the simulations are so realistic, why did they not program a time delay for communication? Such details bother an LII's valued Ti much more than an ILI's unvalued Ti. Ti is a famous detective trait (think BBC's Sherlock or Death Note's L), and the way that Bean collects clues and figures out the broader plan is just like a detective solving a case. Also, he seems more likely to be Se polr than Se suggestive. When Achilles was about to kill Poke, he (perhaps willfully) misunderstood the situation and ran, while an ILI might've been more likely to do something dangerous like calling out to them due to valuing Se but being bad at it. While he has strong Ni, he seems to actually value Ne. When he makes Dimak uncomfortable with his observations about how playing defense in space is hopeless, Bean does not immediately see the one true path of the situation (that they are the Third Invasion), instead keeping open multiple possibilities about what the real point of Battle School is. Additionally, his cautious but thorough exploration of the space station evinces Ne in testing the possibilities of his surroundings.
Bean admires Ender's intelligence and ability, but is never fully intimate with him and never really understands him, always finding him and his leadership skills to be mysterious. This further suggests that Ender is LSI rather than ILI, since Bean as an LII would understand how an ILI works, even if he doesn't agree with him all the time.
Valentine: EIE/IEI
Valentine notes herself that she is skilled at affecting others' emotions to get them to feel/do what she wants. Her letter and visit at the lake are what impels Ender to continue in Battle/Command School. That her interactions with him are successful even though he knows what she's trying to do also provide more evidence that Ender is Fe valuing, not Fi valuing.
She is probably Ni > Si since she has the foresight to collect evidence against Peter over the years, which she is finally able to use to make Peter let Ender go.
She admits to herself that she likes power, suggesting Beta.
Peter: SLE
We don't see a lot of him, especially from his perspective, in Ender's Game or Ender's Shadow. In these two books, all we know is that he likes power and gets it, and that he doesn't find having empathy to be his natural response most of the time. I do want to note that he does show Ender empathy when they thought that the removal of the monitor meant that Ender wasn't going to Battle School--then, Peter said he understood and felt sorry for Ender's disappointment, though Ender was asleep when he said it. SLEs are capable of empathy and are not necessarily "evil"--Peter is just a scary case of an SLE and might actually turn out to be good, considering the title "Shadow of the Hegemon" for a later book suggests that Bean helps Peter at some point.
Poke: EII
Bean chooses her for his plan because he saw that she was the fairest crew boss, who made sure her crew ate even before she did and would help Bean if his idea worked. She refuses to kill Achilles even when she could have, even while Bean was urging her to. She dies because she trusts Achilles too much and because she wanted to protect Bean. Her dedication to fairness suggests Fi lead. Her inability to assert her presence and strength as a crew boss, and general lack of attention to them, suggests vulnerable Se over creative Se, making her EII rather than ESI.
Achilles: SEE
He does not come up with his own long-term plans, but readily, masterfully exploits the opportunities given to him. In the chapter from his perspective in Ender's Shadow, he notes that his murders always required improvisation, but he didn't mind that--he is confident in his ability to improvise on the spot and adapt to situations as they present themselves. This confidence is actually what allows Bean to win. Achilles follows Bean into the duct, Bean's territory, because he thinks it is a good chance to off Bean and is confident that he can handle whatever novelties in the environment that pop up. He doesn't stop to think that Bean might be planning a trap, suggesting low Ni, considering it was obvious back in Rotterdam that Bean was extremely intelligent and strategic.
Achilles doesn't even care when he's kicked out of Battle School, still confident that the universe will bend in his favor, giving him more opportunities to seize. He does not create these opportunities himself, but is simply expert at identifying and taking advantage of them--e.g., the Soviets eventually free him from the mental institution during the war on Earth after the defeat of the Formics. Achilles did not orchestrate this, but will obviously use his freedom for his own benefit in the future in the series.
Back in Rotterdam when Achilles first executed Bean's plot with Poke's gang, Achilles loudly ranted at the line of older orphans waiting for the soup kitchen, and this assertive, almost unhinged demonstration successfully intimidated the older orphans despite Achilles' bad leg and former status as the weakest of the older orphans. All this points to Se.
Finally, he is probably Fi over Ti, since he justifies his murders as restoring moral balance in the universe. Despite his forceful personality, he is sensitive and has his pride easily wounded, which is the main drive behind most of his murders.
Nikolai: ESE
The Socionist in me wants Nikolai and Bean to be duals. But it does make sense anyway. Nikolai is intelligent and able, but not ambitious, so doesn't seem Se/Ni-valuing. When Bean invites him to join his weird special squad within Ender's army, Nikolai demurs, wishing to keep up with his regular toon duties and classroom studies instead of participating in some odd creative endeavor.
His ability to get Bean to open up despite all his walls suggests an Ethical type. His physical defense of Bean as a little brother against a larger kid, before he even knew they are really brothers, is Gryffindor-like, which has an ESE vibe.

