If someone thanks me in Latin, how should I respond? I have been taught to reply sodes, but L&S does not seem to mention such use at all. Did the Romans have any idiom for replying to "thank you"? A direct translation of the English "you are welcome" is unlikely to work, and I don't want anything as heavy as gratias tibi ago, quod gratias mihi egisti. Is sodes fine, but only a modern option?
This is not a phrase which demands a response, at least not from the available evidence. Most instances of gratias ago do not have the thanked person respond at all, and I could only really find one exception.
In Cicero's De Oratore 2.268, we get this exchange:
Arguta etiam significatio est, cum parva re et saepe verbo res obscura et latens inlustratur; ut, cum C. Fabricio P. Cornelius, homo, ut existimabatur, avarus et furax, sed egregie fortis et bonus imperator, gratias ageret, quod se homo inimicus consulem fecisset, bello praesertim magno et gravi "nihil est, quod mihi gratias agas," inquit "si malui compilari quam venire".
There is also frequently acuteness shown, when something obscure and not commonly known is illustrated by a slight circumstance, and often by a single word; as when Publius Cornelius, a man, as was suspected, of a covetous and rapacious disposition, but of great courage and an able commander, thanked Caius Fabricius for having, though he was his enemy, made him consul, especially during a difficult and important war, You have no reason to thank me, returned Fabricius, if I had rather be pillaged than sold for a slave.
It's not a simple nihil est, but I think it gets the point across. But in this exchange, the thanked one minimized the actions, as if there actually was no reason to be thanked to begin with. This is different from the de nadas and de riens that essentially mean, "It wasn't very difficult to do, but I appreciate the thanks anyway."
In Terence's Phormio, we get this exchange:
Nausistrata: Phormio, at ego ecastor posthac tibi quod potero, quae voles faciamque et dicam.Phormio: Benigne dici'. Na.: Pol meritumst tuom.
Nausistrata: Phormio, I by Castor henceforth I will do and speak for you whatever I can, whatever you wish. Phormio: Thanks! (Lit. You speak kindly.) Na. By Pollux, you earned it.
Here it might help to recognize that there were different ways of thanking someone, and this comes after kind words. I imagine it being similar to the English response, "You're too kind!" It's not really the formal thanks and it's not really an automatic "you're welcome."
There might be others, but there's certainly nothing consistent. Searching all possibilities is time consuming, but I think it speaks volumes that it's not something that is easily found (unlike the many ways of thanking someone); despite giving several ways of saying "Thanks!", Goodwin Beach's article "De Sermone Cotidiano" doesn't mention "You're welcome" in any way, though it might help remembering that you're welcome in its current form as a response to thank you was first attested only in 1907.1
I searched for gratias and benigne both, and while responses are often given, there isn't anything on par with English you're welcome, French de rien, or German bitte.
For modern speakers, I indeed have heard nihil est, libet, precare/precamini, and flocci all used, though you should always keep in mind just what you're being thanked for and its appropriate response. Even in English, for someone being publicly thanked for the service to the state, "you're welcome" is hardly appropriate.
1: Though certainly there were other responses, they just weren't "you're welcome."
In @C.M.Weimer's answer we see an exchange of the following kind:
A thanks B for a compliment or a display of esteem/affection
B replies with Meritum est tuum, maybe embellished with a preceding interjecton like pol
I propose a general formula to cover (all?) other cases, provided a reply is appropriate in the first place:
A thanks B for some kind of help or advice
B replies with Gaudeo quod tibi profuerim or Gaudeo me tibi profuisse "Glad I was helpful (to you)"
There don't seem to be instances of these phrases in classical sources, but a quick Google search shows that numerous Latin grammar books have reported them from early 19th century on, so I would use them casually.