No, you do NOT need to run separate wires for the 20A circuit.
The panel comes pre-wired inside; the 20A and 30A circuits share the same incoming power supply.
You only need to connect one set of main wires (line/neutral/ground) to the box.
Are there two separate input wires sized in accordance for each breaker/receptacle or a single 50 amp capable circuit, then split it up on a rail inside enclosure