I'm writing a blues song. There are a few 2-bar sections in the verse where there are no vocal parts and I want the piano to improvise around a specific notated part (it's kind of, but not exactly, an answer phrase to the preceding vocal line). So, say, I've got a sequence of three chords and I want to say "in place of the second chord, you could also use this chord" or "before the third chord, you can add the following fill-in note" or "this lead-up can be replaced with a triplet" and so on. How do I do this, or is it assumed that the performer will improvise around the notated part without being explicitly told "improvise here using the following variations"?
Sounds to me like you should just write out what you want to hear. Improvising by definition is "in the moment composition". In the "old" days, I did a lot of studio work and the composers would either want to hear something specific (which is what I am hearing you say) or want to hear whatever I came up with.
If you give the player tips on what notes to play it essentially handcuffs the process of improvisation. I remember a session once where the writer wanted me to end a solo on a particular note. I would get into the solo and invariably screw up at the end because the realization of ending on a particular note was like being waken up from a dream or "taken out of the moment" so to speak.
If you have a good idea of what you want, why not just write it out. Nothing wrong with doing it that way.
There's plenty of good material here, but I didn't see this mentioned, so pardon me if I didn't catch it in some other answer.
If you really need to hint the players to this extent, I'd suggest that you use cue notes, and perhaps an expression to say what you want ("echo", "answer", or similar) for the first of your two-bar phrases. That's a strategy I would recommend if you're writing for students, who might not be as experienced with jazz/blues as the pros are.
For succeeding repetitions, "etc." or "ad lib as before" would probably suffice.
I think you either notate what you want or allow them to choose.
I would just write slash bars with chord symbols and a written instruction for what you want (e.g. "fill lightly behind vocal" or "improvise freely"). Then trust your musicians ;-)
You can write "suggested fills" but that just begs the question - do you know what you want or not? Quincy Jones says "always leave some space for God to walk through the room". Sounds like good advice to me.