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The Marauders all took pranking very seriously. Even before they perfected their Animagus transformations, they had all learned to imitate each others' handwriting, could even do passable imitations of each other's voices.
And the thing with the secret exchange of secret-keepers? That was the prank of a lifetime, with their lives on the line, and Lily's, and that of the baby they all adored. (Yes, even Peter was fond of Harry; he just loved his own life rather more.)
The one and only time Peter ever passed on the secret by speaking it out loud was when he gave it to the Dark Lord. Every other time, it was a note in handwriting indistinguishable from Sirius'.
Bathilda Bagshot was almost entirely blind at that point, but they didn't let that deter them. The note ended up being enlarged to the size of a small mattress, backlit and inked deeper for contrast, but she still read it, in the end.
It was tricky, sometimes, to avoid circumstances where Sirius was in the same room as a fellow Order member who needed to know the secret, and it would have made no sense to do anything but say it out loud, had the secret really been his to give away. If he hadn't spoken much in the meeting up to that point, then a sudden-onset sore throat did the job. So much medically-unnecessary pepperup consumption.
There were at least two occasions, though, where Sirius had to fake a sudden crisis to get away, if he'd previously been speaking at full volume, or had already admitted to being in the best of health. He saw Moody scowl at his mention of a 'family emergency', saw Alice roll her eyes at Frank when he made his leather trousers split at the seams and departed urgently via the floo to Twillfit and Tatting's, but he hadn't realised the consequences.
The Marauders had been so thorough, thinking themselves so clever, that absolutely everyone had been fooled. Even Dumbledore. He'd cast the initial spell, and he'd had no idea that they'd broken it almost immediately, with Lily re-casting it; Sirius got a lecture about leaving party invitations with valuable information lying around, even if it was in an Order safehouse and even if baby Harry's first smile was a celebration-worthy occasion, but there was no hint that Dumbledore had noticed the location of the party was something he had previously - briefly - forgotten.
So much for information security.
Who wouldn't laugh hysterically, at the realisation of having been so very comprehensively hoist by one's own petard?