If this is a stupid question, please don't mind me. But I spent some time trying to find the answer but I couldn't get anything solid. Maybe this is a hardware question, but I figured I'd try here first. 
Does Serial Communication only work one to one? The reason this came up is because I had an arduino board listening for communication on its serial port. I had a python script feed bytes to the port as well. However, whenever I opened up the arduino's serial monitor, the connection with the python script failed. The serial monitor also connects to the serial port for communication for its little text input field.
So what's the deal? Does serial communication only work between a single client and a single server? Is there a way to get multiple clients writing to the server? I appreciate your suggestions.
Edit:
I forgot about RS-485, which 'jdr5ca' was smart enough to recommend.  My explanation below is restricted to RS-232, the more "garden variety" serial port.  As 'jdr5ca' points out, RS-485 is a much better alternative for the described problem.
Original:
To expand on zmo's answer a bit, it is possible to share serial at the hardware level, and it has been done before, but it is rarely done in practice.
Likewise, at the software driver level, it is again theoretically possible to share, but you run into similar problems as the hardware level, i.e. how to "share" the link to prevent collisions, etc.
A "typical" setup would be two serial (hardware) devices attached to each other 1:1.  Each would run a single software process that would manage sending/receiving data on the link.  
If it is desired to share the serial link amongst multiple processes (on either side), the software process that manages the link would also need to manage passing the received data to each reading process (keeping track of which data each process had read) and also arbitrate which sending process gets access to the link during "writes".
If there are multiple read/write processes on each end of the link, the handshaking/coordination of all this gets deep as some sort of meta-signaling arrangement may be needed to coordinate the comms between the process on each end.
Either a real mess or a fun challenge, depending on your needs and how you view such things.
Serial Communication one to one