No, Lightning messages are entirely private unless the sender or the receiver choses to make them public.
Fountain does that on the sending side.
Splits make it a bit more complicated because anyone on the split can see messages.
I technically make boosts and streaming payments to a Hive address public if you look at the @v4vapp account. I hope to have a fancy dashboard up and running but only for the actual Hive recipient at some point in the next few months.
i know that public and forever is a problem for a lot of people so i was not sure are lightning messages forever :) because listening to podcasting 2.0 talking about cross apps comments and how hard it is is painful :) while we are doing just that right now with Engage and peakd :)