The earliest I remember thinking of myself as having a self was probably around 7 or 8? It was a really weird feeling, this realization.
I didn't really think of other people as having thoughts and feelings of their own until...well, it wasn't one moment but a series of psychological shocks when confronted with the evidence over time. I remember at six saying something to someone and she said something that was completely unexpectedly different from what I would think, and in fact I had no idea anyone could think in that way. I ran into this several times over the years, and even into my 20s.
I know I have had a tendency to react to people as if they're not precisely people, and it always surprises me when they assert that yes they are people with free will. This does not happen nearly as often as it used to (actually, hardly at all, the last time I really remember was in my late 20s). That's not to say I didn't realize that people are people with their own free will, thoughts, and feelings before that, but it's one of those things that's different in the moment versus intellectual realization, too.
I do know I still have trouble predicting and knowing how people will react to things I say, and I've still managed to cross the line in that regard several times just in the past few years. Two cases I specifically remember right now were instances where I badly hurt my niece's feelings because I didn't realize she wanted to talk to me after she'd been in a fight with housemates, and I needed to talk to the housemates about something - so I essentially ignored her and spoke to the housemates. When she explained it, it made sense, but I didn't even think about that.
The other instance was friendly teasing in my WoW guild during a raid. I got a bit too mean with someone and hurt his feelings, and his reaction was a total surprise to me - I didn't really think that he would take what I said that seriously, even though when people said similar things to me, I did take them seriously and ask them to not go there.
So aside from the first time I thought of myself as having/being a "self" I don't really have any specific times or moments where I shifted from one kind of awareness to another. I can sometimes keep things in mind so I know to be careful about what I am writing or (less often) saying, but more often I just don't think about those things and get unexpected reactions.
I do know I get better at it as I get older and I think I got about as good as I've ever been at it about 10 years ago or so.
One thing too is I've played a lot of roleplaying games, live-action and tabletop, and one big thing in that is "don't metagame." Basically, you have to maintain an intellectual theory of mind about your character as well as other people's characters, so you know whether or not your character knows something that you do know. I think this sort of thing may have been helpful over the years, to some extent, but it was also confusing. I had to know what my character knows about what that other character has done, as played by another person. It was usually better to just not know in the first place.
(btw, live-action roleplaying? No matter how much it was related to my interest, I think it contributed a lot to my first serious social burnout - too many people, even in heavily scripted and formalized situations governed by lots of rules).