For the next 20 years, as far as 3PO knows, he is the property of Captain Antilles, doing protocol duties on a diplomatic transport. He is vaguely aware of the existence of the princess but doesn't know much about her. Wherever 3PO goes, being as loud and obvious as he always is, his unobtrusive little counterpart goes with him. 3PO is R2's front man. Wherever they land, R2 is passing messages between rebel sympathisers and sizing up governments as potential rebel recruits - both by personal contact and by hacking into their networks. He passes his recommendations on to Organa.Link (via JWZ)
Yoda is out of the picture by this stage, using the Force-infused swamps of Dagobah to hide himself from Vader and the Emperor. Or something. He is meditating on the future and keeping in touch with Obi-Wan via the ghost of Qui-Gon Jin, which as comm systems go has the virtue of being untappable. Obi-Wan, on Tattoine, keeps in touch with Bail Organa and the other Rebel leaders by courier, of which more later.
As Star Wars opens, R2 is rushing the Death Star plans to the Rebellion. R2, not Leia. The plans are always in R2. What Leia puts into him in the early scene is only her own holographic message to Kenobi. Leia's own mission, as she says in the holographic message, is to pick up Obi-Wan and take him to Alderaan - or so she thinks. Actually, her father just wants her to meet Kenobi, which up to this point she never has. There's a reason for that.