It’s really not that hard. The most common way is to book a cheap hotel room near the donor and then meet them for a long chat in the bar. If you like them enough, you can ask them to donate that night and give them the key for your room and let them know where you have left the sterile container (usually in the bathroom). Then the donor should do his business, come to meet you downstairs to return the key and then leave as quickly as possible.
I think the insemination process is simply to visually inspect the sample to ensure it has liquefied and is not still in a jelly-like state, suck it into a sterile syringe and squirt it into the vagina as far as you can without causing injury. Then some ladies try to achieve orgasm, but I have heard this does not necessarily help. All do agree that keeping the pelvis raised or at least remaining horizontal is preferable to immediately getting up for a walk, as a lot of the semen will likely flow back. I think that is the reason for it being initially congealed for about 10 minutes, so it sticks inside and doesn’t immediately run back out again.
If you are not sure about the donor on the initial meeting, ask them to give you some time to think about the donation process and say you will let them know tomorrow if you want to proceed. The most important thing is to listen to your instincts, female intuition is very evolved and is ignored at your peril!
I personally have all the paperwork from a clinic I am donating to which tested for all STIs, Cystic Fobrosis, chromosome karyotyping, blood group and also checked my family medical history. I think this is the minimum I would expect to see if I were a recipient.