Is there a phrase to use when you suggest someone being on first name terms? I know the verb is biriyle senli benli olmak or biriyle samimi olmak but how is it suggested in a civilized way?

senli benli olmak:

Benimle hemen senli benli oldu.  This is not always a good behavior in Turkish. So "senli benli olmak" is not always a good thing.

samimi olmak:

Onunla çok samimiyiz.

If we want to talk openly: We say: Samimi olmam gerekirse....

But if someone insists to call you siz even though you think it is not necessary.

Btw, I think I found it:

                       Lütfen bana sen de.

Eksi sözlük calls it "Türkçeye özel sizli bizli tabirlerden kurtulma çabası repliği."

or you may say "bana sen diye hitap edebilirsin"

hitap etmek like addressing someone

 "bana sen diye hitap edebilirsin" is OK, if you do not necessarily want reciprocality.

Especially between peers, where you generally do seek reciprocality however, your intentions will be clearer if you say " Birbirimize sen diye hitap edebilirmiyiz ?".

"samimi" is closer to "sincere" and/or "intimate", than it is to "open", as far as conversations go.

"To talk openly", translated into Turkish language would therefore be "açık konuşmak", not "samimi konuşmak". (you dont have to be "samimi" with some one, if you wish to express yourself openly)