Surviving only on another's exhales?
Ok, I'll bite. Now, I'm going to assume that in this scenario, Alice and Bob are stuck underwater. Alice has a SCUBA tank filled with compressed air (not pure O2) and though she won't share it with Bob, she magnanimously agrees to share her exhaled breaths with him.

Exhaled air contains 13.6–16.0% oxygen and 4.0–5.3% carbon dioxide in addition to other gasses which we'll ignore here. The real question here is: can you survive on that? 



13.6–16.0% oxygen is 66%-80% of the amount of oxygen in normal air (which is ~20.1% O2). If we assume that the air pressure of the exhalation is the same as the air pressure at sea level, then the amount of oxygen available to Bob is roughly equivalent to being at an altitude of 2000 - 3500 meters above sea level, which is certainly survivable.

The CO2 is likely to be a bigger problem. CO2 is toxic and, in sufficient quantities, can case asphyxiation even in the presence of sufficient O2. Amounts above 1% can cause drowsiness, headaches, and mild narcosis while higher levels lead to dizziness, confusion, and unconsciousness.

It looks like you need amounts around 7%-10% for death, so Bob will probably survive in the short term, he definitely won't be happy and he won't be doing much. According to wikipedia's page on Hypercapnia, the time one can expect to perform useful activity varies greatly based on the exact amount of CO2:


1 week at 4%,
8 hours at 4.5%,
4 hours at 5%, and
1 hour at 5.5%


If Alice and Bob are just relaxing and waiting for rescue, it's likely that the CO2 in her exhalations will be on the low end of that range and Bob will likely be able to survive (albeit in great distress) for a significant period. If they were exerting themselves, however, Bob wouldn't do well at all.