In almost every case my opinion is the humane technique is to take the animal to a person trained in humane techniques. That is: a veterinarian. I just took my rat (with advanced mammary tumor) to be euthanized. $60 including cremation. That's a small price to pay for her to go peacefully with an overdose or anesthetic rather than whatever I might try.
If people are experienced in humane techniques for whatever reason, and have access to the appropriate tool or drugs, then fine. But otherwise, leave it to the experts. The only animals I euthanise myself are fish and I use legally purchased anesthetic finquel (overdose). As a general rule I don't think anyone should use a method not approved by the AVMA Guidelines.
Also various laws apply. In most US states you can kill you own animal if you do it humanely, but killing any other person's animal may be illegal.
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