Electrical – Dimmer – 120V in, 70V out, normal

Sounds to me like the white wire is the ungrounded (hot) conductor, and should be reidentified as such using a bit of black tape/paint/marker. Power comes to the first switch via the white, and then to the second with a white jumper.

The red wire is the switched "hot", which is used to control one light. The black wire is the other switched "hot", and is used to control the other light. The bare wire is ground.

To install two new switches:

1. Turn the power off, and verify that it's off.
2. Mark the white wire with a bit of black tape/paint/marker.
3. Connect the white wire with two short bits of black wire (pigtails), using a twist-on wire connector.
4. Connect one black pigtail to one switch terminal, and the other to one of the switch terminals of the other switch.
5. Connect the red wire to the other terminal of one switch.
6. Connect the black wire to the other terminal of the other switch.
7. Using bare copper or green pigtails, connect a ground wire to the green screw on each switch, and the box if it's metal.

When done, it should look something like this.

The simplest thing to do is to turn the breaker on and figure out which black wire is supplying the current. (This is arguably one of the more dangerous approaches, but it is safe if you follow precautions like have someone guarding the wires so they are not touched while the power is on.) Use a multimeter and test the voltage between the ground wire and every other wire. Exactly one black wire should have 120 volts on it.

If not, stop and report back here.

Once you identify which black wire is hot, turn the breaker off.

Then wire nut all of the whites together. Wire the "hot" black wire you identified to one other black wire. Turn the breaker on. Note if the ceiling light came on. If not, turn the breaker off, hook up the other black wire to the hot black wire and turn the breaker on. Note if the ceiling light came on. Turn off the breaker.

Now you know: 1. which black wire is the "hot" wire, and 2. which black wire goes to the ceiling fixture, and 3. your "other" black wire, which probably provides power to another switch or outlet.

You should wirenut the "hot" wire and the "other" wire together with a short scrap of black wire to create a pigtail. Connect the pigtail to the switch. Connect the black fixture wire to the other side of the switch. And your whites should already be wire nutted together from before. So just hook up the ground wire to the switch, and you're done!