Each switch powers one or the other leg at all times. You generally need two people working together answer these questions.

Why Would Both Switches On A 3 Way Switch Fail Simultaneously
Why 3 way switch not working. Next remove both switches. Three way switches work by alternating the power connection between 2 legs called travelers. Sometimes one of the switches has become defective. Toggle each switch until the light come on. There are a pair of wires that run between the two switches. To answer questions 1 and 2.
Now go the other switch location. Sometimes a 3 way circuit doesnt work because someone tried to replace a defective switch and did not properly connect the wires. There are single pole switches controlling things from a single location three way switches controlling lights or outlets from two locations and for larger homes or complicated situations a four way switch may be used to control lighting or outlets from three or more locations. The steps i am going to describe may not be the most time efficient way to troubleshoot a 3 way circuit. In this case the traveler wire and load wire were swapped and repairing the circuit involved switching them back. The following method will address both problems at once.
Post person a at the first 3 way box and person b with. First kill the circuit breaker and be sure there is no power to the circuit. At the switch where you found power in step 2 put the other 2 wires you removed from the switch together and wire nut them. There are many ways that a three way switch can be wired but usually power or the light fixture connect to the load side of the switch and traveler wires run between traveler screws on both switches. For the second 3 way box which two of the three wires attached to the original switch come from the other 3 way. Touch the probes to the 3 wires removed from the switch until you get a reading on the tester indicating continuity.
Set the electrical tester to the continuity or ohms setting. Usually when a 3 way switch fails lights can be toggled on and off at one switch but not the other. In the latter one of the switches has likely gone bad. If one of the two 3 way switches toggles the light s on and off the other 3 way switch has probably failed. In the former case you may simply have a miswiring issue. What is the wiring pattern of your 3 way switches.
Follow these steps to find which switch is bad. One with the meter on resistance one tapping pairs of wires together.