Surely 95% of the online bookmakers available on the market are trustworthy. They are checked by international organizations and they can not afford not to pay the winnings (as long as the bettor plays fair) for they risk closure.
Likewise, when you send the documents to be checked you need not worry, for there were no cases reported in which the data of a customer was sent somewhere else. If this thing would happen to someone and a complaint would be made, the bookmaker would receive a more than generous fine and they will never do this thing again.
The only things you need to do is to be extremely careful to the links you access when receiving e-mails from bookmakers. There were cases in which one would receive an e-mail from a bookmaker that said to access the link in order to get hold on an amount of money won, all he had to do is confirm the banking data. Even though the link is placed under the name of the bookmaker, it can redirect you towards a fake website (with a similar graphic) and once you introduce your data, the account will surely be emptied.
It is advised that when you receive these kind of e-mails to access the official website of the bookmaker, directly from the browser. It must be mentioned that the bookmakers have nothing to do with these online fraud attempts.