I don't think there's any such thing as an acceptable amount of time to create a prototype. The amount of passion you have for the idea and the other parts of your life that take your time will feed into how long it will take to create a prototype.
I would say a prototype and a private beta can be synonymous but it is more likely that a prototype would develop into something more substantial which you would then look to publicise and get more feedback from with a private beta.
A prototype is more likely to be along the lines of something you would show to relevant folk on a narrower basis than a private beta, to gauge whether they would be interested in the product.