I can't do much more than agree with Kelly. Especially outside couple relationships, it's often much less likely than people think that people will have a negative reaction to your being trans.
And I think that for "neighbour"/"neighbor", it goes:
-Neighbor in the US
-Neighbour (preferred) or neighbor (accepted) in Canada
-Neighbour in the UK and Australia
As for telling her... No, I don't think people have a right to know they're interacting with a transsexual. If that were the case, transsexuals wouldn't exist; they'd all be transgendered/non-conforming people. Because a transsexual, by definition, wants to be a woman (for a MTF), not someone who wants to be a woman.
Personally, I think you shouldn't lie, but shouldn't say it out of the blue either. To me, going out and say "I'm a transsexual" without a reason is roughly equivalent to saying "oh, by the way, I was born with a malformation and had a tail as a baby". Resulting in a reaction of fearing you as a weird person, not because you're trans, but because you said you were. "Why on Earth did you want to tell me about your tail?" It's roughly the same.
And even if that doesn't happen, it would sound a bit like saying one of your shortcomings as the first thing on a date or job interview. "Before you decide whether you want to be my lover, you should know that I am clumsy when kissing." The heck? Well, you get the message.
Now, after the relationship develops, if a situation arises where either you really feel it's appropriate to say it or you would have to lie not to say it, tell away, but I don't think you should unless you need to, to some degree.
...Lol, Tessa stole my place with a much more concise interpretation.