You will see 12V on both side of an injector without a ground pulse, being provided.

Voltage is a funny thing, it will go all the way to the end of a conductor through as many conductors as it has to. The coil in the injector is a conductor, so the 12 volts provided by the key on, will travel from the ignition switch to the injector, pass through the injector and all the way to the ECM, if you were to measure there. This is normal.

You can see the same thing with light bulbs. Connect one side to 12V+, measure on the other side and you will see the same 12V (occasionally a small drop due to the resistance of the filiment), and all the way to the end of the wire that you will attach to a ground. Then attach the ground wire to ground, you will see the 12V disappear, and the light bulb light up. Again, completly normal.

As far as the issue at hand, I agree, check over the fuse(s) and power going to the ECM, grounds are as equally important, so check them as well.