Yes, at some point you're going to have a connection to, basically, two other characters - the one who is writing on your character sheet and the one you're writing on. It's part of creating the character identity that's at the root of FATE games. This is not a game like DnD where you roll and/or assign a bunch of values to your character and then kind of leave it at that. The creation process includes the whole group in part because the whole group interacts with it.
In any case, in terms of private stuff, there's not going to be a lot on your character sheet anyway at that point (actually, there's a separate "character creation" sheet that gets used for that purpose). There's really not that much in FATE that makes the finished product, actually. There's your name, there's your aspects (which get determined by the above), there's your skills, and that's more or less it. There's also a stress/consequences track that replaces hit points (sort of) and in Achtung! Cthulhu a sanity tracker, but these start out the same or nearly the same for everybody.
FATE is a much different roleplaying experience than most anything else out there that isn't a straight up story game. You may want to take a look at that second video I put up, as it gets into the mechanics, such as they are, pretty well I think. If there's a basic idea behind the game, it's that your character (or someone else in the group) can do virtually anything if you have the FATE points to do it and can craft a narrative around it. There's a complaint out there involving a blind sniper, that basically you can, with multiple invokes and a lot of creativity, make a blind character snipe someone from half a mile away if you've got the points to spend. Personally I see that as a feature and not a bug (you really, really have to have all your ducks lined up in a row to do that, that's a situation that basically the entire party has to not only be in on but willing to "break the bank" to make happen, and if a character is actually *blind* and that's an aspect the GM can just compel them to miss the shot regardless), but it does at least illustrate one thing: this is not a game for power-gamers.