We often have the neighborhood kids come over if we are outside....they just come in the yard and ask if they can join us, sometimes if it is just me in the yard! I've had neighborhood kids help me garden, put up holiday decorations, and do yard sales, in addition to the times they come over to play with my son. Most of the time I don't mind at all, especially because our son is still young enough we keep a pretty close rein on him, and this gives him a big opportunity to play with the other children. It is a compliment to me that they enjoy our company and feel comfortable being at our home. I also think part of it is just our location....we are on the corner and our "backyard" is actually on the side of the house and in full view of most of the kid's homes so their parents can feel comfortable with them being there because they can still see them.
I have, however, been trying to teach my son that he is not allowed to invite himself anywhere...usually he asks if he can go to a friend's house he knows from school, and my answer is always "if xxxx's mom calls me and asks, you can't just go somewhere unless you are invited."
I guess it is just more informal within the neighborhood, especially because we are outside alot. I would feel a little different about a child just asking to spend the night, but on the other hand, I would wonder if they had a bad home life and maybe just felt secure in my home.