Out of 2067 Private Members' Bills between 1983 and 2002 only 256 were successful! This was largely because Government Bills took up the vast majority of the time available in Parliament.
Interactive Democracy treats Private Members' Bills and Government Bills in the same way, in that they both require support from voters to go forward. However, Party Politics would also come into play, as those parties with the ability to mobilise support from their membership, would likely carry the day. The real difference in the balance of power would be because the whips would have far less influence than they do today. They would have little power over party members and none over the rest of us. The result would be that ideas with popular support could emerge from anywhere, with equal opportunity.