How would the barycenter of a double-planet system affect the two planets' surface gravity? Assuming two similarly sized, tidally locked planets orbit a common barycenter, how would the gravity of the barycenter affect the surface gravity? For example, on the inner sides of the planets, what would be the effect of the multiple gravity centers (the planets' and the barycenter), especially to a being on the surface?
There's basically no gravitational pull towards the barycenter.  The two objects are in free-fall around each other and objects in free-fall experience (almost) no gravity from the object they are in free-fall around.  Anyone standing on either planet would basically only feel the gravitation of that object alone, as if it were sitting still in space (though nothing actually "sits still" in space), everything in space is flying through it's local gravitational field.

There can be small variations to the local objects gravity, generated by any rotation.  Tidal locking might be common with a dual-planet system and Tidal locking is usually a relatively slow rotation unless the objects are very close, but while it's a slow rotation, it's still greater than zero, which creates a tiny tidal bulge and a very slight centrifugal force which would effectively lower the gravity.   This has nothing to do with the barycener, it is a product of the objects rotation.

As the objects get closer there is a tidal bulging which has some effect on gravity.   Parts of the planet that bulge outwards experience lower gravity and parts that are effectively stretched in closer to the center experience an increase in gravity.   This effect is pretty small and it's an indirect effect, caused by tidal forces from the other object, not caused by the barycenter.

At the Roche Limit, gravity is effectively cancelled out by the tidal force and objects can drift off the surface both on the near and far side of the smaller of the two tightly orbiting objects. 

The amount of gravitational variation would be small, unless the orbit was very close.

I'd like to post a map of the change in gravity on Pluto's Moon Charon, or Jupiter's innermost Moon Io due to the tidal forces and tidal bulging, but I couldn't find any written results on that and I'm reluctant to try to do the math myself.  Even under very strong tidal forces like those two, the effect is likely still quite small, perhaps in the range of 1% or less.

Here's a question where the tidal force and it's effect on Phobos' gravity is discussed.  You need very strong tidal forces before the gravity gets significantly effected.