I prepare the SSSPs, but I work with the Project Managers / Foreman on the specific details, hazards on site etc. Myself, Project Managers and Foreman all sign off on the SSSP before it gets stored on site. Same process for our JSEAs

Really depends on the set up for the company, but generally speaking i have found that SSSP's done in conjuntion work the best. I.e. there will be parts that the project guys needs to answer for you and part of it that you will need to have input into. What ever happens both parties need to sign off on it.

I had a project team want me to do the whole thing for them once. I said absolutely I would, but then they would have to do EVERY SINGLE THING that was in there and not complain about any of it.
They decided the "them prep, me approve" model was a much better idea!

We are in Civil construction and the PM's complete the SSSP from a template. I then review/approve.