Not switch reference but...

[vec]

What is it called when a language drops person marking while the verbs have the same subject?

Example:
He-sing and Ø-dance = He sings and he dances.

whereas
He-sing and he-dance = He sings and another guy dances.

Is there a term for this? Does this exist even?

[vec]

No?

[Cedh]

I don't know if something like this exists, but it seems plausible to me. I would personally still call it switch-reference though, with the 3rd person marker as zero for "same subject as before" and non-zero for "new or different subject".

(EDIT: Here's a good article on the subject: "Exploring clause chaining" by Robert A. Dooley.)

[2+3 clusivity]

Your examples just seem to be a nom-acc syntactic pivot -- though its hard to tell given that your example verbs are ambi-transitive in english. Maybe I am under-simplifying.

Are you trying to get a term to distinguish the following? For example:

Example 1:
He-needs and Ø-sees pie = He needs pie and he (i.e. the same) sees pie.

whereas
He-needs pie and (s)he-sees pie = He needs pie and (s)he (i.e. another) sees pie.

Example 2:
He-needs pie and Ø-sees pie = He needs pie and (s)he (i.e. another) sees pie.

whereas
He-needs pie and he-sees pie = He needs pie and he (the same) sees pie.

[vec]

Yes, exactly. Your first quote is the system I'm attempting to research.

[Ketumak]

Sounds like logophoricity to me, as found in a number of West African languages:

https://en.wikipedia.org/wiki/Logophoricity