My beloved Smudge was pts at home (4 years ago this Sat) via an injection straight into his heart - this was 2 hours after the vet had originally been and adminstered the injection into his leg, listened to his breathing, declared him gone and left. I was too distraught at the time to make an official complaint of what I think was gross negligence and I never got an apology, or even a card of condolence.
Anyhow, to anyone having to make that awful, final decision and gift of love, please, please, don't let my horrendous experience put you off.
However, do be aware that if you are present, it isn't always - for want of a better expression, 'text-book'. I now choose to take on elderly cats and have had to go through this twice since, and although it's been a far pleasanter experience (I always hold them close and sing and once the needle goes in it has been almost instantaneous) but if, as in the case of Reggie, they are very poorly, it can take a while to find a vein in the leg that doesn't collapse. This is why (to my understanding) the vet will sometimes need to inject an organ direct instead.