Of course, dogs have many varied dialects, but these are the kinds of things I see dogs saying to each other:
Butt sniff = "I am meeting you! Can I meet you?" This may receive a snarl and snap and bared teeth response, which usually means "I'm not comfortable around new dogs because I have dogtism, give me a minute."
Biting and snarling at another dog's neck while the tail is wagging = "WE ARE PLAYING! THIS IS FUN!"
Barking at dogs which are play biting at each other's necks = "Are you okay? I think you're okay, but I'm not sure, because I'm only a dog."
Dogs also code switch between talking to other dogs and talking to humans. Furthermore, when dogs talk to humans, they generally tailor their vocabulary specifically to the people who they interact with most.
Laying on the back or side and making a "face swipe" pantomime with one paw to their own face = "It is time to pet me."
Going straight to the front door and sitting at noon or 9PM = "It is time for my nap or bedtime. You will let me outside in front to go potty, after which I will eagerly run in, straight into my crate, and wait for my cookies."
Stopping and looking back at you after you let him out the back door = "Are you going to come out? I really want you to come out and play frisbee."