I have never had to kill/pts/euthanize any of my pets (myself) as of yet (thank God). However, one of my family's cats was pts at an emergency vet's office.
Having said all of the above, I also keep tropical fish (fish are quite different than cats/dogs/etc. in terms of care). On several ocassions I have had to kill/euthanize fish for various reasons. Namely, when they have extreme injuries (how does a fish break it's back in an enclosed tank?!) or when they have a severe case of a disease that is untreatable (i.e. a growth that causes them to loose control of their back half, or be unable to swim, etc.). There are no vets in my area that do work on fish, so it's usually up to me to make the best guess in terms of care and/or euthanasia. I do all I can with antibiotics, antifungals, QT tanks, extra oxygen, etc. but, when the fish is obviously suffering and I have no means to treat it, there are only two options: allow it to slowly die while suffering or kill/euthanize it.
When faced with that choice for a fish, I will euthanize it rather than let it suffer a slow painful death. I euthanize them via temperature shock, then putting them (w/ the water) in the freezer to make sure they are dead.
So, what you might ask, does euthanizing a fish have to do with a cat/dog? Well, I would take the same approach w/ my larger pets...First, do all I can via vet care, etc. but, when there is no other option, other than to allow them to slowly suffer w/ no cure or improvement in sight, I would euthanize. I would do it by veteranary means if at all possible, but if I was in a situation where vet care wasn't an option and the animal was in great pain and/or distress, then I would rather kill it quickly than watch it suffer.
Now, would I have the guts to kill the animal myself? Seeing as I don't like the sight of blood and am a "wussy", probably not. I would hope that in the situation, I could overcome my fear and do what was best for the animal. I do not own a gun, so I would either have to use someone elses or go to other means.
As to methods of euthanasia other than firearms, the only possibilities I could think of would be allowing the animal to "bleed out" or breaking their neck. I am not sure if either of these is feasable and both would require far more "hands on" than with a gun, so would be even more difficult to do than shooting your animal.
Other than the fish part, everything above is conjecture and what I "think" I'd do in the situation. I hope that I will never be in that situation to find out, but if I do find myself in that place, I am not sure if I will follow what is listed above or not.
As to those equating killing/euthanizing an animal with killing/euthanizing a person.... and I know this might make me unpopular, but, I do believe there is a marked difference between animals and humans. As a christian, I believe that humans have souls and animals do not and that humans were created and given stewardship over the planet and animals (for better or worse). As stewards, we do have power to decide how to use/care for/etc. animals, but we also have great responsibility to do so wisely and with as much respect as possible (i.e. prevent any suffering possible, practice proper husbandry,etc.). So, as a steward over my cats, I am responsible for their daily care, their medical treatment, and, if necessary, making the decision whether they need to be killed to prevent suffering. I believe that if someone is not a good steward over animals in their care (i.e. abuse, neglect, etc.) that it is a sin and that they will be held responsible for their actions by God.
OT: Just as I am a steward over earth/animals within my care, I believe that God takes stewardship over our lives and, when the time comes, it is not my decision to end/take my own life, nor the life of my mother/father, etc., IMO, that is God's decision. Now, don't get me wrong, I belive in doing everything possible to treat disease and to provide care for people at the end of their lives, but when medical science has been exhausted and they are looking at either artificially prolonging their life (i.e. respirator, etc.) or allowing "nature to take it's course" that the latter is better, provided proper hospice care.
Back to the subject at hand, for those that have had to make this decision and have their pet euthanized at the vet's office, you have my greatest respect. I don't even want to think about the time when I will be forced to make that choice. For those of you who have had to euthanize your pet yourself, that is an even harsher pill to swallow and I applaud you for not only deciding to end your pet's suffering, but for taking the steps necessary yourself. Again, I do not know what I would do in that situation, but I hope I would have the strength to do what is necessary.