For the most part, JSPS postdoctoral applications require the student to be acquainted with the new supervisor (aka the "research host") in advance, and discuss with him/her the research plan thoroughly, before writing the application papers. There are some similar programs such as the Hakubi program, where it is not required to know the research host in advance, but in practice, almost all of the successful applications go to applicants who already know the research host in advance, and have prepared together a research plan.
Most of the time, the student applicant is introduced to the host researcher in Japan through a common acquaintance (ex: the doctoral supervisor of the applicant might already have suitable contacts in Japan and introduces his/her student to them). Even so, it is sometimes possible for the student to directly contact by email a potential host in Japan and propose a research topic. If the research host thinks that the research proposal is interesting, and has enough spare time to do it, then both of them can start to prepare the JSPS application proposal in detail (to be clear, most of the burden of preparing a detailed proposal will fall on the student applicant). If the research host already has a good idea and wants to work with the student, then they might take the initiative and propose a detailed plan, but this is somewhat uncommon.
All of these mutual discussions need to happen months in advance of the submission deadline in order to prepare a suitable or competitive research plan. The submission date written in the JSPS website is the final deadline for the host universities to submit the papers. In reality, both the host researcher and student applicant have to submit the paperwork to the host university at least 1 or 2 months in advance of the JSPS deadline.
The application papers have a section that must be filled out by the student applicant, and another that must be filled by the host researcher. The application must be submitted by the host researcher to the administration office of the host university, that will then submit it to JSPS (however, there is also an online submission system where the student applicant will have to log into and submit some papers). JSPS then investigates all of the submitted documents to make sure that the applicant and host researcher sections are coherent with each other, and whether the plan is feasible as a whole.