Does the Board of Directors make the decision?
Board of Directors inform the SEC about the intent of "Major holder / Promoter / Promoting Bloc" in taking the company private.
Do the existing shareholders get to vote?
Yes / No. The process involves making a too good offer to refuse to all the existing shareholder to part with their shares. Thus say a promoter may increase his stake from 25% to 90%.
If I were a tiny minority shareholder, what kind of process could I expect to see as this effort proceeds? I know eventually, if successful, I would have to relinquish my shares. But along the way, how does the process unfold, how am I affected, and what are my rights? Specifically, how are decisions made, and what voice do I have?
This is different in every situation as there are multiple ways this happens. This is governed by various company laws and SEC rules. Also read Squeeze out.
If the intent of the promoter is to de-list the shares from "Exchange" and thus take it private ... the reasons could be to get more flexibility and stay away from public glare. All the promoter has to do is meet the requirements of de-listing the stock from stock exchange. Each exchange has rules NYSE if the number of shareholder is less than "X" or share in market is less than "Y" etc.
In such cases one can still be a minority share holder; but selling shares etc becomes difficult.
If the intent of the promoter is to gain 100% of the stock; this is slightly different. He may indulge in series of legal manoeuvring to achieve his goal but this would take time.
In all cases, the interest of minority shareholder is generally protected. He would get more than fair value of compensation for the stocks held. What he may loose out is; if he believes this is a great company and I want to hold it for 20 years before I sell ... so the lost opportunity sometimes is not safeguarded.