Peltier cooling only half of one side
If There are a few problems that could be happening:


It's broken by mechanical or electrical stress
The thermal contact is wrong


One thing to do would be to check the resistance at room temperature with another module of the same make, it should be pretty close. Sometimes you can check the ohms of the module with the datasheet. With any module if the impedance is 1k or higher at room temperature, then there is most likely a problem.

Another thing that can kill peltiers is thermal expansion, ceramic (usually aluminum oxide) has a coefficient of thermal expansion, which means the hot side gets longer and the cold side gets shorter, this can break modules.

If the module is behaving differently, the most likely thing is because a physical change has occurred in the module, possibly reducing it's effectiveness. 

Peltiers are fragile, most all peltiers consist of two pieces of ceramic sandwiched with a junctions made from positively or negatively doped bismuth telluride crystal. The junctions are all in series, if one breaks it reduces the efficiency of the whole module. Sudden shock from dropping or other objects can break modules. Peliters can also break by exceeding absolute maximum values listed on the datasheet, this usually happens through thermal stress. Run too much current through a peltier and you can kill it.