No offence [meant]. None taken.

[Theseus]

It means "what I just said may sound like I'm insulting you, but that's not what I intended OR it doesn't apply to you even though it seems like it could" and the second character's response means "Don't worry, I'm not offended."
--A: They're all a bunch of brainless dropouts. No offence.
--B (who is a dropout but not brainless apparently): None taken.

Often, one says "no offence" before saying something with brutal honesty.
--"No offence but you do not play football well enough to stay on this team".
--Joe: No offence, Sally; you've got fat.
--Sally: None taken. I guess I should go on a diet.
-- Pete: Excuse that last remark. I did not want to offend you.
--Tom: It's okay. No offence taken. :huh:
PS In my notebooks I have found:-
Με συγχωρείτε για τη τόλμη--Συγχωρεμένος [!]/ Δεν ήθελα να πειράξω κανένα. Μα δεν πείραξες κανένα. Is that idiomatic? I am trying to find where I got it from.

 

[Themis†]

Καλά τα λέει το notebook. Η πιο κοινή ίσως μορφή:
- Συγγνώμη αλλά...
- Συγχωρεμένος.

 

[dolphink]

Εναλλακτικά, ο διάλογος θα μπορούσε να πάει κάπως έτσι:

Joe: Χωρίς παρεξήγηση, Sally...
Sally: Καμία παρεξήγηση...

 

[drsiebenmal]

Επίσης:

--Με το συμπάθιο... (a little bit outdated)
--Κανένα πρόβλημα.

 

[Theseus]

no offence meant. None taken

I am trying to find this thread but I always find the search tool less than helpful. Can someone help me with this? Thanks!:s

 

[daeman]

...
No offence [meant]. None taken.

For searching within Lexilogia, we type the search term(s) in the box above "Lexilogia search" in the black horizontal bar just above the thread title and hit enter, or click on the magnifying glass icon next to that to open a new Custom Google Search (in Lexilogia) browser tab. That's how I found it, typing in "offence". Don't click on "Lexilogia search", that option hasn't been fixed yet. :)
I'm moving these two posts to that thread, Theseus, OK?