Historically, these mirrors were displayed in dining rooms where entertaining would occur and with the outward bending glass, this allowed the butler to see the entire room (ie. when glasses needed refilling or prevent guests from trying to steal the silver) while staying discreetly out of sight. It was the ultimate domestic surveillance system. I always see a few of these at Scott's Antiques and they are awesome. Now they can be found in living areas above a mantel, in the bedroom, or in the hallway.
Lynn Morgan at House Beautiful
Notice the 13 balls around the mirror. This was to represent the 13 original colonies. Bullseye Convex Mirror from Richard Rothstein
Here is a smaller version painted white from Lonny
The company, Syroco founded in 1890 used resins to replicate the original wooden adornments and carvings on the mirrors to make replicas. These more affordable versions can be found on Ebay and Etsy.
Federal Style Mirror by Syroco found on Etsy from Affordable Adornment
Another Etsy find from Green Zebre
Syroco Federal Bullseye mirror from Belated Designs on Etsy
Convex mirror from Cafiero Select