On a dating site or singles posting "Discreet' means affair.

There are times in real life dating and relationships where discretion happens on purpose.

Discreet
adjective
~ careful and circumspect in one's speech or actions, especially in order to avoid causing offense or to gain an advantage.
~ intentionally unobtrusive.

Discretion
noun
~ the quality of behaving or speaking in such a way as to avoid causing offense or revealing private information.
~ the freedom to decide what should be done in a particular situation.

You don't take your date, lover, friend or spouse everywhere you know because it is inappropriate to the relationship.
You don't say or do things that are inappropriate.

Even people that tell you they have nothing to hide are discreet about some things.
Discretion is not a selling feature for an open relationship.
It is a method of preserving distance, keeping at bay.