How do I transfer all my data from a iPhone 6se to iPhone 12 mini? Do both phones need a SIM card inserted?

The new one needs the SIM card in order to activate and to transfer all data and info from your old iPhone to the new one , first make an iCloud backup then get closer both iPhone and wait for the screen setup and follow all the steps until you see the option to transfer all info to the new iPhone then the camera from the new one will turning on itself to scan the cloud from the the old iPhone, don’t move away both iPhones leave them together on the side until the transfer process finish

No. You could pull out the SIM card from an older iPhone and it should be fine transferring data. As noted, you'll need a working SIM card (plus a working Wi-Fi internet connection) to activate the new iPhone.

There are multiple ways to transfer. I've only done it by restoring a backup from the older device to the newer one via a Mac, but there are other ways like Quick Start and through iCloud. Quick Start only gives one chance after coming out of factory condition, so that can be tricky It might take restoring to factory condition if it's not done.. I know some people charge up iPhones where it wakes up, and I think powering down before starting the setup will mean it will still turn on again in factory condition with the setup pages.