No, quite the opposite unless you are switching the ground through the relay. Now on the stock wiring yes that is the case, so on the relay coil you should see 12 volts on both leads until you press the horn button. If you press the horn button and there is still 12 volts on both then your horn button contacts aren't connecting.
Now on the relay you should have 12 volts coming from the battery, and nothing on either side of the horn. When you press the button you should see 12 volts on both sides of the relay contacts, and 12 volts on the plus connection of the horn. If you still see 12 volts on both sides of the horn with the button pushed then you have a bad ground to the horn.
OK after carefully re reading your post, yes there should be 12 volts on the front horn until you press the button as the stock horn does not use a relay...