Mar,
I am currently in a BSN program and I do not know which accrediting agency services my school. I am under the impression that it doesn't matter who accredits the program. The main objective will be to be able to sit for the NCLEX exam and pass. Once you have done this, you will be awarded the title of RN.
The only caution I would make is to be sure that whatever program you choose is accredited for awarding college level credits. For example, the school I attend is accredited by SACS. The credits I earn are fully transferrable to any other institution. Some private schools are not accredited by this (or similar) entity and although you can earn a degree from these private schools, none of their credits are transferrable for advanced degrees. It would be a shame, IMHO, to pay out the funds required for a degree only to find that if you wanted to persue a degree at a public institution, you would essentially be starting at step 1.
I hope this makes sense, and anyone else is welcome to chime in if I have anything incorrect.
ETNguy