Does laser removal on unwanted Tattoos affect wanted Tattoos nearby?

If the laser doesn't hit the ink in the tattoo, it will be safe. If the tattoos have any overlap or are up against one another, then you can have some collateral since the laser will likely hit the adjacent tattoo. These are easy to correct by going to your tattoo artist for repair. If you have adjacent or overlapping tattoos, then I recommend going to your tattoo artist first to get their opinion regarding possible repair.

No, laser treatment of a tattoo would not affect surrounding tattoos not being treated.  Treatment occurs only for the actual tattoo being treated.  The main function of the laser is to break down the ink particles to smaller sizes in order for your own body's immune system to resorb it.  Consequently, untreated tattoos are not affected.  

When treating a tattoo, the laser is very focused and should not affect nearby tattoos. It is easy to tell exactly where the laser hit because of the change in color of the tattoo. If it is not hit at the time of the treatment, it won't be affected later on either.