If one kidney fails, does the other weaken?

Not necessarily. If one kidney fails, the other kidney (if healthy) will compensate by increasing the blood flow (GFR) through it. If one kidney fails completely, the other will have more filtration but it does not necessarily weaken. The caveat is if the process that caused the first kidney to fail persists (such as diabetes, or hypertension), then this could cause the other kidney to fail as well.
Usually. Usually when one kidney fails for some reason, the other kidney will grow in size and take on some of the load of the other failed kidney. This takes time (years). Of course, this all depends on the other kidney being healthy and one not having a disease that will affect the other kidney. Nevertheless, one should follow closely with their doctor to ensure the one remaining kidney stays healthy.