I posted this in another forum, but here are my feelings. Meeting in a neutral site is basically just a vacation and in general vacations are much more wonderful than normal life. What does this accomplish? In my opinion the first meeing could be at a neutral site, with shared expenses, if there is a plan for future visits. I would prefer to meet the woman in her country. To meet the friends and family, see her in her normal surroundings. Seeing someone dealing with their close people can give an insight to the upcoming family life with this person.
I would then have the woman come to me for an extended stay, as this is the closest you can simulate the life you will have together. She can see the city and learn the neighborhoods and such and the two people can experience what a regular life can be like. At this point, both people should have a better idea of how compatible you are and if this is the person you want to marry.