Because an element of my life may become public soon ( nothing notorious or bad in any way ) and so it occurs to me that I would like a separation between public and private. What is already public is difficult to undo but when starting with a clean slate ...
I think the key to this, is keeping seprate identities for your public and private personas.
Start simple, seperate email address. Separate phone number.
You the personal email address and phone number for anything personal/not related to the public persona and vice versa.
I would possibly look at a PO Box or proxy address for receiving mail. This could be used when registering domain names etc..
If you don't have some sort of mail proxy setup, might be worth fitting an external post box - more of a safety thing than a privacy thing.
Then you want to look at what information is already available publicly about you. Social media profiles, directory listings etc.. ... is there anything you can remove? (yes it might be archived somewhere, but archives don't last for ever the advent of GDPR has shortened the life of many.
I guess essentially, you want to keep as much personal information as you can out of the public domain. Particularly information that could be used as a basis for fraud, setting up fake accounts, making false implications etc..
You probably want to take Twitter accounts, facebook, linkedin pages etc. in your public perona's name so there's something official looking and distinguishable from fake accounts people might set up for nefarious/vexatious purposes.
It does really depend on what you are most concerned about. And also whether it is yourself you want to protect, or dependents. It is very wise to consider privacy when taking on a public role.