How can I set up a second hand Apple Watch that was given to me without using the other person apple ID

If the Watch still requests you sign into their Apple ID it means the Watch was not removed from iCloud yet. They need to follow What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support

If you already gave away your Apple Watch, you can erase and turn off Activation Lock from iCloud.com. Here's how:

1. On a computer, go to Find My iPhone on iCloud.com and sign in with your Apple ID. 
2. Select All Devices, then click your Apple Watch.
3. Click Erase Apple Watch. Select Next until the device is erased.
4. Click the X button  next to your Apple Watch.