Is it a problem? Or is the excess water supposed to run to the street?

Maybe this would be a good spot for a dry stream bed - with some sweat equity and a source for rocks. Trying to dam it up in the yard will only create ponding. As for the street - storm runoff routinely goes to street corner/low spot drains.

I have a corner lot with a grated street drain that I keep free of leaves and debris so pollutants don't carry into the system.

The source of the problem is that instead of a nice gentle swale graded between the properties (a joint problem) it's a tiny "v" on one property, concentrating the water into a thin stream. Regrading is the solution, but if you don't want to spend the money (don't blame you) you must establish dense plantings (like sod) within the eroded area. It doesn't help that the stream runs in a shaded place (below the tree) where sod won't grow adequately. For that area, you must find a groundcover plant that will grow low and dense so it is capable of protecting the soil.