How can I make sure my Mac has access to files on its own SMB server? I have a Mac mini that runs a Samba server through the "file sharing" option in settings.

I can create files on it and my clients can access it. However, creating files on the client end causes my Mac not to have access to those files. 

There doesn't seem to be a way to change that on the client end. I tried going into properties and unchecking the "read only" option on the file but it gets checked again when I hit "apply." My clients are Android and Windows.
This is the reason why companies set up shared directories - to allow or deny group read / write permissions to files.

You will want to think about how you wish to have people authenticate and whether all shares can default to everyone can write.

On the Mac - files like this typically are homed in /Users/Shared so you could move those files to a place that’s naturally open to all accounts or change the permissions on the files you chose to share from the sharing panes.

Another last bit of advice, spend a little time thinking if you can use POSIX group / ID based permissions or want to layer on Access Control Lists (abbreviated ACL) that are more granular and can do things like add a group and then deny one member of that group from files and folders.