Is there a formal way to say "from the horse's mouth"?
Information that you get straight from the horse's mouth means information that you get from a reliable source.
Origin - In horse racing circles tips on which horse is a likely winner circulate amongst punters. The most trusted authorities are considered to be those in closest touch with the recent form of the horse, that is, stable lads, trainers etc. The notional 'from the horse's mouth' is supposed to indicate one step better than even that inner circle, that is, the horse itself.
from the horse's mouth - from an authoritative or dependable source, from someone who has the facts.
- "I know it's true! I heard it straight from the horse's mouth!"
- "Are you sure she's leaving? Definitely, I heard it straight from the horse's mouth.
To answer your question, you can say it formally "from an authoritative source" or, not so formally, "from someone who knows the facts".