I think the most important thing is to ensure that the euthanasia will be done in a way you are comfortable with, and then have someone there who reassures the horse. Some horses are 100% okay with the vet and tech, others really seem to be reassured by “their” people only. You know your horse best.
As far as how the euthanasia is carried out, I will absolutely not allow any horse in my care to be euthanized without an IV catheter in place, and lots of sedation beforehand. These are both pretty common place now, but I have seen enough without these in place to make sure that I state it ahead of time. The excitement phase without sedation is awful, as is losing a vein part way through. Sedation does prolong the actual dying process a bit (lower heart rate and blood pressure means it takes a bit longer to circulate the final injection, but the horse is without question quiet and comfortable), but it is kind.
Be aware that the older guys can go down with a thump after sedation, simply due to their being less mobile and agile. A younger horse is more likely to lie down in a more usual fashion. A good vet or handler will help guide them down in either case.
After the final injection, there will be breathing, often irregularly, and sometimes groaning or sighing. This is a normal part of the dying process, and happens even after the heart has stopped.
I don’t say any of this to try to scare anyone off of being there - if handled well, the process is sad, but kind and peaceful. Not knowing what to expect is worse. Not knowing what to advocate for is worse. I am a Critical Care RN in my life outside of horses, and have guided so many through withdrawal of care and watching loved ones die. Most people say that knowing what to expect ahead of time makes it easier.