OK - what do you mean by "private"? Let me clarify terms. There are basically 2 different types of higher education: Traditional and Commercial. Traditional (not for profit) schools can be either "public" (tax supported) or "private" (not tax-supported). Commercial schools are all for-profit organizations - they may be either private (investor owned) or publicly owned (listed on the stock market).
Are you referring to for-profit schools when you say "private"? If so, the answer is "no", they are not as well respected as traditional (not for profit) schools. Their primary mission is to make a profit. They do not provide an environment that attracts quality instructors. They spend more money on recruitment than on faculty salaries.... They cost significantly more than traditional schools.
If you are seeking a BSN, $50,000 is twice what you would pay at a public, non-profit school. You should also investigate accreditation status, prior student complaints, access to federal loans (not possible at many for-profits), and NCLEX pass rates.