(Meant to put this in the comment but hit length limit).
Caveat: I've only written a couple of small dzil plugins myself, and am not very familiar with its internals.
dzil uses Log::Dispatchouli, which is also written by rjbs. Log::Dispatchouli is used by several other projects (including projects started by people other than rjbs), so it's not exactly an internal dzil logging system.
From what I understand, Log::Dispatchouli only has two logging levels, normal and debug. Debug logs are not normally seen in output; you have to enable debugging first. Normal (and debug) logs can be muted if you enable muting.
To enable debugging in dzil, add a -v command-line switch. This does not seem to be documented in 'dzil help' nor 'dzil help COMMAND'. But rjbs blogged about it.
Now to produce logs inside the dzil plugins, you just call $self->log(...) or $self->log_debug(...). The second one will be seen only if user passes -v option.