Of course there is more current draw with a bridge configuration. Each amp has to provide the same amount of current through the load. The voltage can be doubled, but you still need current to provide the power.
When bridging amps, you have to consider the minimum load and double it for reliable operation. If each amp has a minimum load of 4 ohms, then the minimum speaker load should be no less than 8 ohms.
If the amp is capable of stereo as well as bridge operation, then both amps are using the same power supply. That's why there are usually terminal connections instead of jacks for bridge operation - the speaker load must be "floating", meaning no ground connection. Otherwise, watch out for the sparks!
There is a bridge design, I think I remember Crown used it first back in the early 1970s. It was also used by Sunn in their Concert amp series. This design "floats" the power supply instead of the speaker load allowing one side of the speaker load to be grounded.