Here's how I gleaned most of what I know about Saito from, this is almost all taken from the 3 parts of SAC, as seldom other places ever mention him, including the Manga.
When resented with no direct information about a character I glean information from who they are, what role they play, how their attitude is portrayed, their speaking parts (or lack thereof). Even right down to their facial expressions, stances and position when standing in a group. Don't think for a moment that these things aren't considered when the animations are put together, because just the same things apply to the other characters, and in most all cases every character has their own behaviour to fit their character, as you would expect.
There are also parts where the character does participate, and while few and far between with Saito these often give clues to his personality.
My deductions are as follows:
He's pretty quiet. It goes without saying really, as it's probably one of the reasons he doesn't say a lot. You rarely hear him butt in in briefings, or make off-hand comments.
He's a loner, but not because he's a-social or insular, he just works best alone. He doesn't mind working with other people as long as they let him do his thing. The scene he has with Batou in SSS is a pretty cool example of how he works in a partnership. Batou tries at one point to make him suit up in thermoptic camo, but Saito puts him straight pretty fast, because eh knows he hasn't got time - it's his job so he calls the shots. He also has a scene in SAC1 Ep1 where he works with Boma to ambush the think tank. In that instance too he works flawlessly but he's giving the orders. Most of the time on large team ops Saito is on top of a building or on a chopper. He's the team sniper, and he's the best damn shot they have. That usually involves doing the job yourself. Any interference, like that seen in the episode in SAC1 where he tests the 'sniper assist' module and doesn't get on well with it because he's 'too human', just serves to irritate him.
Which brings me to his attitude. He's there to do a job and woe betide you get in his way. His commitment to his unit and his work is unquestionable. He might not show it often but he is as dedicated to Section 9 as every other member.
He's a thinking person. He is usually seen in team briefings stood with his hands in his pockets, leaning with his foot against the wall, taking in everything that is said. The old 'poker face' covers a great analyst, IMHO. He can read most situations like an open book. He rarely says anything though, which is perplexing. Perhaps in his thoughts he figures how everyone else fits into the plan, and factors in his own part. Because he knows every person in Section 9 so well from his working experiences of them, he likely can pretty much see them slot in to the picture without having to ask many questions.
He's a master of on-the-spot thinking. You have to be to do that job. Imagine the war scene you see him operating in in the 'Poker Face' episode. That sort of environment is kill or be killed, you don't have time to think. Also, he and Batou cook up a solution to trap their target in SSS as they are running to the top of a nearby building. Saito orchestrates the operation, thinking on his feet while lining the other guy up for a shot. The scene also shows, when they get to the top of the tower that his ability to assess a situation is calculating but extremely swift. He knows his job inside out, and does it without thinking. A sign of a true pro, as everyone in Section 9 is, but also someone able to think outside the box.
He's not nearly as much of a grouch as he looks. He actually comes across in the scene at the beginning of SSS when he gets back from Africa as a fairly affable and cheerful person. He cracks a fairly happy smile in that scene, in fact. Something of a revelation when you usually only see the Poker Face. He strikes me that he's very good at canning his emotions when it matters I think, and only letting them play when there's no 'recoil'. I have to say I can see him, Pazu and Boma in a bar mulling over their day, having a game of poker and a joke about. The poker face thing he put on with the rookies in 'Poker Face' struck me more as a gag to amuse himself and put the wind up them than a serious attitude. I think a person like Saito probably has a really dry sense of humour
He respects those who show him respect, and he doesn't go back on it. He shows that with his dedication to Section 9 and to Major Kusinagi, who earned it on the battlefield as his adversary, but also by not killing him in cold blood, instead giving him a way out.
That's about all I can drag out of my memory for now. It's not a complete character assassination by any means, but it shows, at least in my mind, many facets of Saito as a person. Some are common to all members of Section 9, but some make Saito who he is. I can identify with a lot of those parts of his personality.
*collapses exhausted*