Thranduil's elves communicate very very differently when they're alone in their realm versus when they have any visitors or leave their realm for any reason. And pretty much nobody other than the Greenwood residents know about it.
Tolkien says that they forgot their old Silvan Language in favor of Sindarian, but I think they just just aggressively leaned into the other races thinking them very simple and stupid and just told them they forgot it. They told the other realms they didn't remember anything other than very basic Silvan words, but secretly use it as their realms main language.
If there's nobody else to listen to them, Silvan is the only language they speak. Most residents are bilingual between Silvan and Sindarian but there's still a few pockets of Greenwoods population that don't even know any Sindarian at all, due to a mixture of spite and it being unnecessary for their daily lives.
 Thranduil and Greenwood do this for a number of reasons, the main one being that protecting and preserving their culture was always hugely important to them, and preserving their language was always one of the most important factors in that. They felt that if the other Elven Realms were to hear their language, learn it, and occasionally use it, they would force a change upon it. And by extension the rest of their culture too. They were worried that what happened to their kin in Lothlorien under the Noldor's rule would happen to them too.Â
They also hid it because as they were watching their lives grow darker and darker, and were growing more wary of those they could trust. And if you don't know who or what you can trust, what's one of the methods of ensuring you can keep yourself, your family, and the rest of your people safe? You speak in a language your enemy or others who might hear, understand, and pass on the knowledge don't know or understand. They can't tell Sauron their plans, their routes through the forest or anything else if it sounds like straight up gibberish. Their messages and communications can't be intercepted if it doesn't sound like a language. It helps to keep everybody safe.Â
On top of actual spoken language in Greenwood, I think the Silvans would use and adapt their forests way of communicating as well.Â
The Silvan Language doesn't have as many specific or descriptive words, because they share their emotions and feelings through their connection with the forest and therefore each other. They don't have a use for having a word for distinguishing between various related but not identical words.Â
Like: Anger, frustration, annoyance, peeved, irked, rage, testy, irritable, etc, because they can feel the difference in the trees.Â
So instead of saying something very specific, they can say more general things like: 'I'm upset." Because the elves around them can feel for themselves in what manner and capacity their upset.
So their connection with the forest is huge in their communication.Â
On top of that I also see them copying or mimicking animal sounds from the forest that they often hear, especially if usually hear it in reaction to something.
Bird calls are the most widely used, especially among the warriors because they're so versatile. They can convey entire intricate plans with different bird calls because each of them means something very specific. Or just tell each other their specific locations, if they see or hear something, or anything else. (Aragorn, since he used to spend some time in Greenwood without his family as a teen was taught several of these bird calls for safety purposes. He and Legolas routinely use the "Where are you?" "I'm over here!" Bird calls when they're in crowds bc it's just easier until the day Aragorn dies.)Â
Other animal sounds that the Silvan elves often see and interact with have been adopted as well, usually without a conscious effort to do so. They tend to come out as an automatic reaction to things:
If you bump into them unexpectedly, they'll make a little 'Mehhehh' sound that Deer do.Â
If you say or do something they don't like, they'll hiss gently or growl.Â
If you surprise but delight them, they'll make little happy chirping sounds or something that almost sounds like a purr.Â
If they get very excited but don't have anything specific to say they'll chatter like squirrels.Â
And any other number of animals sounds. Was originally very alarming and confusing to the fellowship when Legolas chirped for the first time. Aragorn thought it was hilarious because he so used to it, he forgot it was weird.Â