Is it OK to send the same research proposal to several professors?

If you wrote the proposal entirely yourself, I see no problems with trying multiple universities. Do note the deadlines: you don't want to be in the situation that you need to choose between accepting an offer of a "lower-tier" university while still waiting for your first choice university to respond.
If you incorporate the suggestions of the faculty member of one university in your proposal to other universities, I'd at least let that faculty member know. Something like: "your university is my first choice, but as you know it is very competitive. Do you mind if I use this proposal to apply to other universities as a fallback"? This achieves 2 things: first of all it prevents a possible faculty member that might feel taken advantage of by using his/her comments to get into a different university. Second of all the advice you get might predominantly apply to his/her university, and might not generalize to other universities.

You don't say which country/cultural context the applications are for. I'm familiar with the UK setting in particular. Here, there is no problem using the same research proposal for different applications -- it's even expected.

There is the additional wrinkle in the situation you describe that one potential supervisor might be investing significantly in the development of your proposal. If you were to go to a different institution, having made good use of that contact's input, then it would also be reasonable for that contact to be a teensy bit hacked off -- unless there were mitigating circumstances (e.g. appreciable difference in funding levels).

Generally, however, it will be expected by potential supervisors that applicants will be testing the waters at more than one institution (they ought to advise applicants to do this in any case!). Transparency should not be a problem in most cases (i.e., inform your contact that you are, in fact, applying elsewhere too). Unless you're dealing with a prima donna/primo uomo, that shouldn't be an issue.