Creatures with this ability - and it is inbuilt, not learnable - can understand anything said to them, and can make themselves understood to anybody.
When they speak, they seem to the listener to be speaking in whatever language the listener expects to hear, or the language with which the listener is most familiar.
This is nothing like the interworld common languages, which are constructed languages that have to be learned as normal. Demonic language can't be learned.
There is no concept of word sounds in angelspeak: a celestial can understand the meaning of what is said, but not the words themselves. A lot of wordplay is wasted on them, and they don't understand concepts such as rhyming or scansion.
If it was a letter handwritten by someone who poured a lot of angst or joy into it, some psychic traces may remain. If it's printed by machine or a sign painted by a sign-painter who's just doing her job, no chance. Ishtar once stared for ten minutes at the sign to the previous inn before coming to the conclusion that it meant "Surely it's time for my lunch break now"...
Most demons and angels have trouble conceptually with pictures, though they understand the idea of a simple representative symbol (for example: a particular set of stripes on your tunic to signify that you're a celeste penulte).
Anyone, including people with no telepathy and mindblind people/anti-telepaths, can understand demonic 'speech' if they hear it. In fact they have virtually no choice.
What the listener experiences is what they assume to be a person talking in either their mothertongue or in whatever language they were expecting to hear. Unless they have some reason to suspect otherwise and are really concentrating hard...
Just like normal speech, if you can't hear the carrier sound you won't hear the meaning.
It works in the same way through other vectors (eg light, finger-touching, spores; however the target species communicates).
My idea of the rationale behind this is that everyone, when de talks, usually intends to convey information and be understood, and because of that there will be some psychic 'bleed' or 'noise' accompanying the words. (Not as organisedly as demonic 'speech' rides carrier sound, but serving the same purpose.) This bleed is what demons unconsciously pick up and understand.
Exceptions should be suggesting themselves already. For example, if you don't have any intention of conveying a message but just say "blah blah", the demon doesn't understand. Indeed, if you were extraordinarily mentally disciplined you could teach yourself to speak actual words without intending any meaning behind them. This would prevent a demon understanding you while a non-demon could. And if a telepath's shielding works to exclude all psychic noise from escaping, a demon probably wouldn't understand den. (As far as I'm aware, demons can understand a shielded Shaded telepath - but again, might change my mind.)
The unsconsciousness of demonic 'hearing' implies a few more things. For example, if you managed to broadcast telepathically to a demon - see telepathy notes below - it might react as if you'd spoken aloud to it, not being able to tell the difference. (This depends partly on its shiftiness and whether it's paying attention. A genie probably wouldn't be confused.) In addition, most demons can't understand real written and spoken languages even if they want to - see Writing, above.
This is not the case with their hybrid animals, i.e. hellhounds. They think like sapient mass-based animals and can be read like them, although be aware of the bonded hellhounds caveat; there are areas of a bonded hellhound's brain that are inaccessible. A telepath may not notice this unless de is probing deeply or looking for something specific.
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