In terms of biological transfer there are a number of possibilities
Smooth still world with high pressure bridge
The remoteness would be similar to that between America and Europe on our Earth. Airborne seeds, spores, birds and anything that was small enough to hitch a ride on a bird such as insect eggs and bacteria etc would be able to make the journey.
Smooth still world with low pressure bridge
The remoteness would be greater than that between America and Europe. Birds would not be able to survive the low pressure environment, but airborne seeds, spores and bacteria should be able to make the crossing depending on how much weather we allow on “smooth still” world.
Real worlds don’t tend to be smooth or still. They tend to have mountains and weather and both of these would cause problems. The presence of mountains would create also sorts of tiny gravitational anomalies which would help to destabilize the Roche world. But the weather would be the real killer. The presence of Earth sized storms migrating across the planet into an area of low gravity will cause chaos.
In an area with little net gravity, gravitation would not play the leading role. Instead other forces such as storm fronts would. Even at 1g on the Earth the sea can be whipped up to 10’s of meters high and rocks and boulders from the sea floor can be thrown into the air. Such a storm moving into the air bridge area under perhaps just 0.05g would carve up the sea into enormous mountains and the higher the seas got the lower the gravitational attraction would be. It is highly likely that sea water at some level would be transferred. So some lower forms of sea life might also make the journey.
Best guess world
Given the propensity for things that can move freely to end up spinning when a force is applied (like stars, planets and storm systems). I suspect the air bridge area itself would suffer the most horrific perpetual circulating storm which would suck up water and seafloor materials and fling it out into the bridge. Some of the smaller solid and liquid particles could easily be transferred between worlds.
So in summary in best guess world seeds, spores and bacteria for sure. Other microscopic life forms such plankton very probably. Larger life forms such as insects, possibly especially if carried aloft inside a small branch or similar in egg form. Macro life forms such as fish and birds: I doubt it due to the violent turbulence and the unpleasant mixture of mud, water and air that would span large parts of the bridge.
Of course it is entirely likely that something would evolve on one of the worlds to live in such an environment. But that’s another story.