We are talking about publicly traded companies here. You can always participate in the buyback: just sell your shares, they're publicly traded, so there should be no difficulty. It doesn't matter much who's on the other side of your trade, as the arbitrageurs will ensure that the buyback is priced into the spot price.

many companies have multiple types of stock, and only publicly trade a portion of their shares (i.e., not treasury stock) and often only certain classes of their stock. sometimes they trade less than half of the total shares, which is how you can have majority privately-held companies trading shares publicly.

Also, to avoid insider trading issues, companies restrict the times in which shares are allowed to be traded by employees.