The picture of the switch showing the two black screws does look like there is a recessed common bridge as you stated. Read your instructions that came with the switch. They are the hot side and as shown below without a bridge and a common hot wire would normally be wired to both switches. A bridge would not require it.
But in your case it appears that you have two hot wires (black), or the two black wires going to the ceiling light. therefore, you would need to break the bridge if there is one. Get an ohm meter and check if there is any continuity between the two black screws.
What is the brand and model number of this switch?
The pic below is from a google search on
Leviton double 3 way switch diagrams. The wire on the left screws would be your red and white wires. Please note that the white wire in your switch box is NOT a neutral. The neutral is at the ceiling light and it may be feed from the other 3 way switch.
From your pictures and description
(hot on all leads), your double switch in the circuit is the two switches on the left of the diagram.