Yes they are, out of the box.

v3.20 and above will do these:

• while wearing the gag, you cannot speak on chat channel.
• while wearing the gag, you cannot make outgoing IMs...
  • except you can IM your owner and up to three friends**, at "garble off" and "garble loose"* settings, being as your owners and close friends** should still be able to understand your mumblings at these settings.
    
  • at "garble tight" settings*, even your owners and friends** cannot understand your mumbling.
  • note that incoming IMs are not blocked by the gags. Why? Because they're gags, not ear plugs, that's why! Sheesh.
• if you are locked, both the HUD and gag are not detachable.

The only way to be able to speak on chat, or use IM without restrictions is to either (1) remove the gag, or (2) use a regular SL Viewer (where chat is garbled when spoken on /8 but IMs are allowed).
Now, if you ungag and still cannot chat or send IMs, generally it's not Lulu's Signature gag that is stopping you - you have some other collar or item that has issued the restrictions commands (except if you have HUD v4.12.02 or below and Gag v3.21.05 or below - there is a known issue).

If all else fails, you might need to relog to clear the RL Viewer's memory.

* If you are Initiate, you will only have the "garble on" or "garble off" choices. "Garble on" is the same as "garble tight" in this case.
   Once you reach Novice, you will have the additional "garble loose" setting.
** the IM exception list (up to three friends) can only be set by the sub's owner (she can be self-owned though, to set this herself). Just  say "/1.imexception" or hit "IMexcept" button on HUD menu ("/1.") to add up to friends.