How to hide paragraph items based on permission
Field permissions is not what you want, that only does permissions for fields, not for the entire entity like you want.

There's a hook called hook_ENTITY_TYPE_access, you can use it as hook_paragraphs_item_access(), that would let you alter permissions (Look at the paragraph_bundle_permissions submodule for an example).

As others have said paragraph items don't actually have the owner on them - neither does field collection. They aren't really mean't to be separated like this. You can add the property yourself if you want to.

I recommend changing the Paragraphs to not be paragraphs and be something like Nodes instead. If they have permissions and are owned by someone, paragraphs aren't really the right thing. You can use Inline Entity Form if you want to embed a form for it. Since they are nodes you should be able to have more control on permissions.