Second toilet runs after first one flushed?
Sounds like a problem unrelated to the flapper. The "running" you hear is likely the fill valve. Perhaps the fluctuation of your water supply pressure when one toilet's fill valve opens is causing the other toilet fill valve to momentarily leak-by or open. I have seen this happen before and a rebuild of the fill valve washers (or fill valve replacement) should fix it.

Try shutting off the supply to the suspect fill valve and open it up. Inspect the valve washers for debris, lime, scale, mineral deposits. Flush the valve by cracking open the water supply while it is disassembled. If the valve washers are worn or soft (if you get black stuff on your fingers when touching them) they need replacing.