They might use telepathy with their eyes.
When an alien locks eyes with another alien, they can freely communicate with each other until the bond is broken. You can manage that using a physical link (made of light?) generated from glands directly underneath both aliens' eyes. This also makes it easy to have aliens overhear conversations/partake in group conversations. They can make eye contact with the link, and they link with it too, being able to again hear and speak in the conversation.
Another similar idea: You can manage the bond using an invisible link that both aliens sense somehow. This way, no
bodyalien can listen in on conversations they shouldn't.
A third idea, and my personal favorite: Combine the two ideas!
The alien initiating the conversation is able to consciously choose what type of connection they make with the person they are talking with. Both ways have their advantages and disadvantages, but the second one is usually more desirable, as privacy tends to be better. Maybe the second option costs more energy/brainpower/whatever to initiate?
TL;DR: Aliens communicate telepathically. 👽