It is perfectly legal for them to give you a jumpseat in the cabin. You cannot ride in the cockpit jumpseat these days. In the cabin you are just a passenger who didn't have to pay.
As far as what your experiences will be like? Depends on a bunch of things.
If you used to work for that US airline, or have friends there, were furloughed from there, former ALPA member etc those things will help in your getting the jumpseat. Don't know anybody there, ex military and not ex US air carrier pilot, not US citizen, etc those things may not help.
Your way of asking, if you brought the guy a new Wall Street Journal etc can all make a difference. In other words "this may work, but your mileage may vary.....
Basically it is up to the Captain and he is doing you a favour as we can't reciprocate. Would I count on it to get to work? Not without a backup plan. Will it work? Sometimes.
Some airlines are better than others. US Airways is usually nice to other carriers. AA I understand you can't get past the gate agents to even ask the Captain. If you ride on the commuters or anywhere where the guys are looking for a better job, you might find yourself welcomed as royalty.
So, plan to have an ID 90 with you and if you get the jumpseat it is a good day. If not you use your ID 90. Biggest problem will be that all the airplanes are full, which may make you consider buying real tickets especially if it is near the holidays. OR move and quit commuting by air.