Freedom of speech is protection from government making laws that restrict your speech. Reddit, twitter and such are not government and not performing government functions. They may kick you out of their place for what you say, don't say, or just because your name contains a character that they don't want to display.
But no, being banned from a service for speech does not incur any need to inform others of your ban. It's between you and the service.
On the other hand, IP-Bans are a measure that is usually temporary in nature due to the nature of shifting IPs, and often come together with account-bans, and account-creation-bans on the credentials blocked accounts had.
Example: you get banned as user TESTTHING from TESTTHING@TESTTHING.TEST, then any account using the same e-mail address gets banned and new accounts attempting to use the same mail address are denied or get auto-banned, together with a ban of the new IP. If TESTTHING uses their sockpuppet from a currently banned IP the usual result is, that the sockpuppet and its credentials go blacklisted too.