I think I get it, so when you said "shell out some cash to keep your exposure to the company's upside" you mean buy the options for the value you were granted them (the "strike price" I think). Is that right?
Well, you own the options after they vest; you exercise options in order to purchase shares. In other words, you own the right to buy shares at a specific price (that's the option), and exercising that right means actually buying those shares (generally for a price far below the current share price). You can then hold on to the shares, or turn around and sell them for an immediate profit (depending on the terms of various agreements).
Careful: you probably own the options only for a very limited time after you leave the company. You cannot generally hold on to your options until the company is (say) certain to be acquired; you generally have to exercise them almost immediately after leaving.
Unrestricted common stock you can keep long-term. Options, not so much.
Another thing to be careful with, if you leave the company pre-IPO or purchase, you need to exercise those options, but the money you are spending is still very much a pie-in-the-sky bet. Until the shares become publicly traded, you can't do anything with those shares, sell them, trade them, etc. You can only hang on to them in the hopes they become worth something someday.