I deeply appreciate your kindness at this stage of your dogs' life.
Because I worked in an animal shelter for more than 5 years, I know how many people cut and run at the end of their dogs' lives. They drop them off at the local animal shelter, "surrendering" them, and carelessly drive away, relying on the organization to handle ending the life of their disposable pet -- with some measure of mercy. I saw dozens and dozens of good old dogs frightened and confused at the very end of their lives, and staff members who were severely demoralized by the requirements of their jobs.
It takes moral courage and responsibility to be there at the ending of your pet's life with dignity.
You will know whether to do this at home or the vet's office. The office option is not always the bad one, especially if there are family members who don't want to accompany the pet into the procedure room and some who do.
Dogs are a wonder of creation, IMO. Thank you again for being a responsible pet owner in a nation that all too often ignores the old, the ill and those who cannot speak for themselves.