While private, NYCOM does have a preference for in-state students. They don't, however, exclude all the very qualified out-of-staters from walking their campus.
Because you're interviewing relatively early for an out-of-stater (NYCOM doesn't usually interview out-of-state students until later in the application season), you are probably highly desired by the NYCOM AdCom. Way to go!
As for where you'll stay, I can't say for sure. When I interviewed there I came from home in the city, so it wasn't much of a trek. There are several hotels/motels in the area that are reasonably close, though I've heard the accomodations are horrible. Call NYCOM regarding lodging for the night before the interview. Staying with students may be a possibility, but bear in mind that NYCOM does not own any dormitories and so any students you might be staying with will be off-campus as well. It's probably better to stay in a hotel/motel anyway since if you stay in a student's room, you'll probably end up on the couch or the floor.
The accpetance:interview ratio for NYCOM is pretty high according to the AdCom people. They give all the students on interview day a quick debriefing, and I distinctly remember that the Director of Admissions said they tend to take a lot of the people they interview. That's good news for you!
Good luck on the interview. Remeber that they require you to fill out a secondary on interview day, so bring along a copy of your most up-to-date resume/CV. You will also be asked to write an essay on why you would like to attend NYCOM for med school.
Tim of New York City.