Update: I didn't comment on the fact that speaking about oneself can also be a matter of using "he" or "she" for oneself, as well as using one's own name.
I have never heard anyone in Japanese use
彼女【かのじょ】(she) to refer to themselves. As far as I can tell, it has more or less the same implications that it would in English, and you can follow the same instincts. So, for example, you might use the third person when writing a bio on a web site when it's understood you're writing about yourself.
The point is that using he or she has no special place in Japanese for referring to yourself.
Using one's own name to refer to oneself, however, is something that is done in Japanese and different from English.
Women up to around 30 or so use their own name in place of a first person pronoun commonly enough. However, it is reserved for certain social contexts, like within their relationship, family, or close friends, and definitely not for circumstances like work or school.
Some people think of the use of one's own name as being "girlish", but that is debatable. I think it's a matter of opinion as well a context.
Men can also use the term, but in even a far, far more restricted sense. A father talking to his young child, or a man in a very specific context with his girlfriend/wife/partner.
For men learning Japanese, I would absolutely recommend not ever using one's own name in place of
俺【おれ】. If the circumstance comes up when it might work, as it has with me in some relationships, you will know beyond a shadow of a doubt. Or, put another way, if there is a shadow of doubt, you shouldn't do it. (Note I'm not saying it's an indication of closeness, just that it's the result of a context far too specific and complicated to describe here.)
For women learning Japanese, you would have the option to try it, but it's an art, not a science. You would have to watch your female peers as a guide to see if they felt comfortable using it given their age, standing, and social context.
For both men and women learning Japanese, it is far more beneficial to learn how to drop the first person pronoun altogether than it is to master using one's own name for that purpose.