You could use an Uno (or another mega 328 based board), although it has only 1 hardware serial port so one PC could use that and the other would use a software serial port with a TTL to USB adapter (FTDI). Software serial ports are slow, but if you can live with less than 38400 baud rate that will work.
A Mega has 4 hardware serial ports so you have high speed on all ports. Only one port is USB so the TTL to USB adapter would be needed for the other port.
You need an application on each PC to receive and send the serial data. The IDE serial monitor can e used for very simple stuff. A terminal program ( TerraTerm, RealTerm) can be for more complex stuff like writing and saving files. Or custom apps that connect with other PC software.
A good primer on serial communication is the serial input basics tutorial.