It was in 3.x, but not anymore
The Bluff skill from the previous edition had the exact case you’re asking about:
Delivering a Secret Message
You can use Bluff to get a message across to another character without others understanding it.
However, the paradigm was shifted in 5e. You don’t “use Deception” anymore. Instead, players are supposed to describe what their characters do. Now it’s the DM who is in charge when determining the outcome, instead of just asking for the same opposed roll every time.
That’s it — the DM uses existing described checks as models for improvising others, she doesn’t need confirmation for making rulings. This is the principle known as “rulings over rules”.
5th edition chooses not to explicitly codify many things. This is especially important for “doubletalk” since there’re many factors involved — you and your allies might know each other well, you can share knowledge which may help to convey information, you can use gestures and cues in some situations, et cetera.
You can definitely try though
The closest thing which can be found in the 5e rules is the Thieves’ Cant feature — it does exactly what you’ve described but is a specific class feature, not available to everyone.
While Thieves’ Cant gives a guaranteed success through using a secret code, other players can always try to do the same thing using their convenient language:
The only limits to the actions you can attempt are your imagination and your character’s ability scores.
When you describe an action not detailed elsewhere in the rules, the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.
PHB p. 193
According to the rules, the DM has to determine, if the action is possible and what ability check is needed (if any). It is the DM’s job as described by the source books:
The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.
PHB p. 174
Questions like “what ability check is needed”, “do allies get automatic advantage”, etc., can’t be strictly answered here, because all there things depend on the particular DM in this case.
For example, the DMG explicitly allows not using dice at all:
One approach is to use dice as rarely as possible. Some DMs use them only during combat, and determine success or failure as they like in other situations.
With this approach, the DM decides whether an action or a plan succeeds or fails based on how well the players make their case, how thorough or creative they are, or other factors.
DMG p. 236 The Role of Dice
As a player, you describe your actions or roleplay the dialog, and the DM determines the outcome, with or without an ability check.