I come from a family of artists - most of my aunts and siblings are either writers, drawers, painters and/or musicians. My mother is one of them and really is gifted when it comes to choosing the right words.
When my father died, my mother wrote the eulogy for his funeral. And what she wrote truly shook all of us, be it relatives or old coworkers. It was moving for our family because we don't talk about feelings and she somehow managed to write something that truly represented my father without using cliché phrasings and for his old friends because well, since he didn't talk about feelings, I suspect they didn't know that side of him.
Both relatives and acquaintances kindly asked my mother whether they could have a copy of the eulogy. Some did it very politely, some others were quite inconsiderate. Here is some advice based on both experiences:
- explain why you'd like to have a copy. Briefly recall the relationship you had with the deceased (in case they don't know you so well) and tell them that you think the text could be a great reminder of what kind of person they were.
My uncle and grandparents used something along the lines of
What you wrote was really moving and truly represented who he was. It'd be really nice if you could hand me a copy of the text some time.
No pressure on when to give that copy, no hard feelings if my mother wanted to keep it for herself. They explained how it'd benefit them to have the eulogy.
You could add "I think it'd truly help honoring his memory" at the end of that but be careful only to do so if you were really close to the deceased and know that the eulogy wasn't embellished for the acquaintances in the audience. It'd ruin the legitimacy of your request.
don't ask the speaker right after the funeral. They're mourning after all, now probably isn't a good time. Reach out to them a few days after the ceremony (I'd suggest one to two weeks so that it's still "fresh" ; coming with such a request months after would simply seem weird), preferably via email so that you don't take the chance of making the person sob on the phone - it'd just be embarrassing for both of you.
if you weren't that close to the deceased, it might not be a good idea to do so. As you said, a eulogy usually contains some intimate details about their life and it would sound creepy for someone they don't know that well to want to keep a record of this private information.
don't insist if they seem uncomfortable or initially refuse your request. It'd just seem like a pushy and inappropriate behaviour to have in times of emotional distress and is likely to burn bridges between you.