Obviously this is only true at agencies with an assistant, and then only some of them, but at every agency I've worked at, it's gone something like this:
(1) My boss asks me to read a partial she requested because she's especially busy lining up interviews for her bestselling author.
(2) I read the partial. (Obvious)
(3) We have a 2-minute conversation, tops, about the partial. Sometimes it doesn't take two minutes. I once worked at an agency where for the most part, the agent did not personalize the response to partials unless they were very close, but she did have a special form letter response specifically for partials. My current boss responds herself, in her own handwriting (not mine).
Most of the time with partials, I don't say to reject unless it's obviously so horrible that the query was just plain misleading. Partials are something the agent was interested in from the get-go, so I'm not quick to dismiss the work, and if I do, I have to give a reason. (Occasionally "The author can't spell" is enough) Here are some common responses I'll give her:
"I'm not thrilled about it, but to be honest, it's not my type of literature anyway, and there's nothing wrong with it on a sentence level, so you might like it."
"The author has a specific style of prose. You'll decide whether you love it or hate it in two pages."
"It's very similar to a lot of stuff on your list in terms of content, but I don't think it's as good as any of the stuff on your list."
"I can't make heads or tails of it. Was this an e-query you responded to? OK, you look at it then. I have no idea."
"Does this guy know you, or something? Why was this requested?"
"I hate literary fiction, and this is literary fiction, and I actually liked it a little, so that's pretty much a stunning recommendation."
And so on. If it's a reject and she doesn't want to read it herself for time reasons, she asks me specifics about it so she can be more personal in her reply.