I think I've solved it:
Suppose the gods sit around a table (or in a circle).
Furthermore, we assume the gods are well-mannered beings; in particular, they will refrain from answering if not asked.
We now ask:
"Oh, gods of Truth and Falsehood, pray tell me at once if the god at your left-hand side will answer "no" in your language to my question?"
Argument:
1. Due to a good upbringing, the God of Random remains silent.
He is therefore readily identified.
2 Case 1. God of Truth has the God of Random on his left side.
The God of Truth is obliged to answer "no" in his language, since otherwise he would have implied that GoR says something in response to our question, i.e, he would have uttered a lie.
Hence, GoF is forced to say "no" as well, in order to deny the truth that the god on his left side (GoT) will say "no".
Conclusion:
In this arrangement, we'll get identical answers from the two gods.
3. Case 2. God of Falsehood has the God of Random on his left side.
GoF is obliged to answer "yes"; otherwise, he could be charged with having implied the truth that GoR won't say anything at all.
Hence, GoT must say "no" in response to our question.
Conclusion:
In this arrangement, we'll get different answers from the two gods.
4.
Hence, we may conclude:
If both answer identically, the GoT sits at GoR's right-hand side, and the word said means "no".
If their answer differs, the GoF sits at GoR's right-hand side, and his answer means "yes"