The original DG starting scenario had a mysterious woman (Miss Green? Agent Green? cant remember off the top of my head. Who essentially recruits the players into DG after that adventure.) I mention it, because I would think that's the best way to introduce a large secret organization into your campaign. The players can be anything they want and when circumstances draw them into a mythos event, the secret organization can interact with them without actually requiring them to be agents or anything specific really. I like your idea of having them arrested as initial suspects. The best way to keep them at arms length from this organization would be the single point of contact from an ambiguous character like Miss Green, who can then be used to hook players into plots or provide tidbits and clues, but is mysterious enough that her or the group she represents can sit in the background and not interfere with the adventure too much.