"Know thyself": from the first Socratic dialogues on, that has been the commandment of western philosophy: I must know myself: which means that I occupy both the subjective and the objective position, standing outside of myself to be both the knower and the thing known. I don't, however, think that babies learn about themselves this way: before being able to know themselves as themselves, they know how others see them, how others act on them.
I've been thinking recently, for instance, in a piece of contemporary child-rearing advice from the United States: when a child does well on a test or another academic endeavor, we shouldn't compliment her as being "smart" but as "working hard" or "being dedicated." The idea, I think, is that intelligence is innate, but children develop persistence and dedication, so parents should focus on the virtue that can be trained and improved.
The problem is partly that intelligence isn't innate, but largely defined by others; I've worked with kids living on the street, forever defined as retarded in their school records, whose minds challenges me much more than any of my colleagues from grad school at Harvard. Even more important, however, is others seeing you as smart: once you have that label, people listen to you more, they laugh at your jokes when you're a kid, they push you into intellectual pursuits, they read your words with more care. And in the process, the smart kid actually becomes smarter; she trains her mind to do well what people consider to be smart. (There is, by the way, pretty decent evidence that the climbing IQ scores (30 points higher across the scale since 1900) aren't as much about changes in the test, as they are about urbanization and modernity. Our lives have taught us to think in new ways, ways that are rewarded by the test. If IQ tests measured ability to predict weather or sense when a mountain lion might attack you, we'd have dropped even more that we gained.)
Those last paragraphs are the practical upshot of the fact that a baby knows herself as others know her: we have to be very careful how we know her. But there is also a strange philosophical conclusion to this process: even as we become older, we still have to know ourselves through some metaphor of the same process. I try to look at myself through other eyes, see myself as others see me. Rita and I see Helena, and define her in that process, but she also sees us. For instance, I have always considered myself a good cook, but Helena doesn't much like the food I prepare. She prefers Rita's. I find myself thinking of myself in different ways, trusting myself less in the kitchen. At the same time, she finds me much more reliable and trustworthy than I thought I would be with a baby. I always thought I would drop her, but since she never expressed fear that I would, I also came to trust myself.