a building's plaza, an empty lot, or a common area
First thing to know: you may not need a permit!
If your project is a smaller event, or a simple intervention like a garden that requires no structural changes, you will likely not need a permit.
But you will need permission.
Where to start:
Find out who owns and/or manages the property. Make sure it's truly private.
Use the Miami-Dade Property Appraiser's Office website to do a property search. Once you locate and select the property on the map, a window will magically appear with owner information. Unfortunately, you won't find contact details (except mailing address) on this site.
If this is a building that you commonly visit, contact the property or facility manager's office.
If this is an empty lot (which may have seemed like a vacant park), you'll need to reach out to the property owner and/or real estate firm that works with them. Google is now your ally - start your search with the owner information from the Miami-Dade Property Appraiser's Office. Warning: You may have to resort to snail mail.
Set up a meeting, and explain what your project is.
It is really helpful to bring along a picture.
Because these are private places, there is no standard process and all permissions are at the property's discretion. You can review the information on the Parks page for a general idea of how they handle permitting.
Even with the building's permission, your project may need additional permits. For example: large events or anything involving construction. The property's management should give you an idea of what's needed.
Get it in writing
Let's assume you acquire the necessary permission (and additional permits, if necessary). Draft an MOU (memorandum of understanding, sample here) between you and the property that clearly defines your roles, responsibilities, fees, and any additional information.
Get it done!
There's no stopping you now. Your project (and you) will likely bend and twist a bit as you go, but just stick to it and get it done!