That's a tough decision. I would say do what your heart tells you. I once had a cat that lived 19 years before we had to put her down. My twin sis chose to go and be with our cat and I stayed behind. I'm still to this day happy with my decision because I can still picture my cat in her healthy, happy years. On a flip side, my twin sis was happy she chose to be with her in the end. Seriously, listen to what your heart tells you... Good luck!
My husband and I both were with our dog when we had to put him down. He was in my arms and I whispered in his ear how much he was loved and what a wonderful companion he had been. My voice was the last thing he heard and he knew he was loved. My heart was broken but I owed it to him to not let him go alone.