As Barberry is an invasive species, we hesitate to recommend you keep the plants and treat them, but in order to try to diagnose the phenomenon, additional photos would be useful. Are you able to take a closer, in-focus picture of the nodules? Can you also cut one off, with a little bit of the normal stem on either side, and slice it in half lengthwise? (Or cut off two nodules and cut the other cross-wise.) We're looking for indications of how the wood inside the swollen area is structured, as this may be the work of a pathogen.

If you need to submit more than three photos, you can reply more than once to attach other files.

Normally, such a disfigurement should be pruned off and disposed of and affected plants tend to recover if it's not a systemic infection. In this case, though, the overall decline of the plant may be coupled with other environmental stressors, such as over-watering or poor drainage. Since Barberry is so resilient overall, root rot from wet soils is one of the only conditions that harm them.

This may be a good opportunity regardless of the cause to replace the Barberry with a different non-invasive shrub. We'll see if we recognize the symptoms as a pest or pathogen that could impact other plants, though, once we can view the additional photos. The neighbor's Barberry may be a different cultivar that has more natural resistance to this condition, or may simply not have been contaminated yet if a pathogen is at work.

Miri