The 18-year-old Academy graduate penned his first professional deal with the Cherries in April and has since gone on to make four substitute appearances for the club's first team, the most recent of which came in late September.
Daggers boss John Still said: "Josh is a player I have watched a few times and when I first saw him I had the Matt Ritchie feeling.
"He was enthusiastic, bright and lively. He can play wide, centre midfield and off a forward. We have been helped by AFC Bournemouth financially and it hasn't cost us any money to bring him here.
"I think he will be well received by Daggers supporters because of the enthusiasm he shows."