Does my MagSafe Duo charger need to be plugged in to charge my watch and/or phone

It needs to be plugged in to work. In addition to the other information in How to use MagSafe Duo Charger with your iPhone and Apple Watch - Apple Support be sure to note the following:

It's important to plug into a power source before placing your iPhone on the MagSafe Duo Charger. This allows MagSafe to verify that it's safe to deliver maximum power. If you happen to place your iPhone on the MagSafe Duo Charger before plugging into a power source, remove your iPhone from the MagSafe Duo Charger, wait three seconds, and then put it back on to resume maximum power delivery.

and

All power adapters have different ratings for the amount and rate of power delivery. The MagSafe Duo Charger requires the following ratings to deliver faster wireless charging:

• USB-C connector (USB-A is not supported)
• 9V/2.22A power adapter provides up to 11W of power
• 9V/3A and higher power adapter provides up to 14W of power
• iPhone 12 mini and iPhone 13 mini can get up to 12W for faster wireless charging with at least 9V/2.62A
• Higher wattage adapters at or above 9V/3A will also deliver a maximum of up to 14W peak power to your iPhone

To use both the iPhone and Watch charger at the same time, the MagSafe Duo Charger requires at least 15W (5V/3A or 9V/1.67A), but this will result in slower charging.

When Lightning accessories such as headphones are connected, charging is limited to 7.5W to comply with regulatory standards.

Yes. I have the MagSafe Charger (non Duo) and leave it plugged in. It doesn't draw a measurable amount of power when not in use according to my meter.