The problem is they're on the same circuit, and they may actually be overloading that circuit. The classic example is a gaming PC (10A) or server (13A) and a laser printer (7-12A in operation, 50A when cycling on the fuser assembly). In case you're counting, circuits are typically 15 or 20 amps. So you overload them right quick. The circuit breaker is a slow-trip, and might take 15 minutes to trip at a modest overload.
However, the overload is happening on the same 2 wires -- and the resistance of those wires (against the high current) is causing an excessive voltage drop.
Voltage Drop = Current * Wire Resistance.
The resistance of a wire is proportional to its length, so if it's a long wire run back to the panel, it will make the voltage drop worse.
Another thing that will make voltage drop worse is poor connections in receptacles or switches. This is common with "back-stab" connections and can happen with screws also. Feel for any outlets on the circuit getting warm. Outlets shouldn't get warm.
Running another circuit from the panel will solve the problem.
It will not matter if the circuit is on the same pole or the opposite pole - the problem is the long wire run. So the laser printer will suffer its own voltage drop in its cable run, and the computer will feel only its own voltage drop in its cable run. If the computer does feel it, that indicates a supply problem - too small a subpanel, or a developing problem in the service drop from the street.
You could consider a multi-wire branch circuit - however that could cause interaction because now you have 3 resistors in play instead of just 2. MWBCs are generally more trouble than they're worth, especially now that AFCI is required and GFCI is widely used.