You didn't mention which branch but at least for AFROTC, the norm is you take classes at the your school and take part in activities at the host school. At some larger crosstown schools the cadre may visit for a class. Contact the school you are considering for your specific situation.
Yeah, there are two types. One is where you literally have to enroll in the ROTC at the other college and nearly every single thing you do is at that school, and the other is where the other school is just the host school and the ROTC personnel will come to your college and teach you there. You will just fall under the other school's PMS and unit.
It depends on the schools too. When I was at Univ. of WI, we had a "cross-town" with UW-Whitewater for AFROTC. Our cadre would make the drive down there one night a week and they ran their own wing for PT, etc. When I was a student at UW-Milwaukee, they had a cross-town with Marquette for Army ROTC. The kids from UWM would literally take the city busses for an hour to get over to Marquette.
