In general people don't run programs if they don't want people to participate!
Of course it depends on what the program is, how it is funded, who the other partners are and what language or grade requirements there are.
I realize that all schools and research facilities are different, however, If you are talking about a study abroad program for school, not necessarily for research, then I think you don't really have to worry so much about being accepted per say.
For instance at my school they offer various types of studying abroad, we have our own campuses abroad in 2 countries, as well as direct exchange programs set up in maybe 8 or so countries. Then there are sponsored programs by groups like IES (Institute for International Education of Students) in all sorts of locations.
The school you attend probably has a set of requirements that you need to meet in order to be eligible, like at least sophomore standing, cumulative g.p.a of 2.8 or something of that nature. Then you would talk to your schools study abroad counselors or advisers about which program you are interested in. They should be able to walk you through it. If its a direct exchange program then you may need to fill out the application for that school somewhat like a transfer student, but again that will differ for each school.
I would say that in terms of being accepted into the program, if you meet your school's requirements then you should be fine if there is enough room. However if you are planning on studying at the most prestigious school in whatever country you choose, then perhaps they will be more picky about who they allow to study there.
One final note would be that if your school can't offer much help on the issue, then try talking directly to the school you want to study at, you may find that many schools are more than welcoming to foreign exchange students.