For spectre v1 and v2, right now (on existing hardware) mostly nothing separates threads from processes. In the future, process isolation is a good candidate for designing hardware + system software such that different processes are isolated (via partitioning the caches, etc).
You probably still want threads within a process to share cache hits.
In terms of the possibility of exploit, as I understand there isn't at this point any isolation between processes.
In terms of the ease of exploit, being able to run untrusted code in the same process as the victim helps quite a bit. Otherwise, you have to find a gadget (i.e. qualifying bounds check for v1, indirect branch for v2) in the victim process that you can exploit from the attacker process. Possible, but quite a bit harder than making your own gadget.
This all ignores the forward looking reasons process isolation is a good idea. I can't keep track of the latest mitigations in Linux, but they pretty much all will only help between processes by flushing various hardware data structures. And hopefully someday we will have hardware actually designed to restore the guarantees of isolation between processes.
I'm pretty sure this is accurate, but I'm just a random guy on the internet so don't trust my word for it too much.