I'm ashamed to admit I was surprised someone had the videophone figured out as early as 1910. I also need to apologize to that old crazy guy in the park—your Prohibition-era webcam stories may have been true after all!
OK, it's a sketch of a concept for what the French thought videotelephony would look like in 2000, not a working videophone, but still it shows people were thinking big at the time.
In fact, even earlier in 1878 a wily inventor named George du Maurier actually published a conceptual upgrade to the era's "speaking tubes" using this drawing below, which depicts a "viewing display" to go along with that generation's literal series of tubes.
Only in 1927, with the help of IBM, would the traditionally accepted view of a "videophone" come to pass. The screen displayed at brisk 18fps and was run using one those room-sized computers (Ed. Note: as correctly stated in the comments, this was not necessarily a "computer" but a half-room full of necessary equipment for the broadcast). The video was one-way, but the audio allowed then Commerce Secretary Herbert Hoover, seated in D.C., to speak with an audience in New York City. [Wikipedia - Thanks, Blam!]