I just bought a 3-way switch. I notice that when hot wire is connected
to the black screw, both traveler wires are always hot no matter how I
flip the switch (the hot wire coming out of the other 3-way switch is
always hot, not responding to the position of this switch).

I measured the voltage across the black and either of the two traveler
screws and it was 0. However, if I measure the resistance, it does seem
to work (0 between the black and one of the traveler screws and infinite
if the switch is flipped. Same for both traveler screws.)

This doesn't make sense, does it?