The situation is actually worse than what twincitizen described. Minnesota voters, along with voters in most of the southern states, do not have referendum or ballot initiative rights. There is an extremely limited scenario when Minnesota citizens can chime in: legislatively-referred constitutional amendments. But voters only get to vote after it the amendment has been approved by both chambers of the state legislature.This is how it happened in Colorado though. The referendum was for an amendment to the Colorado constitution that provided for personal use, sale and cultivation.
One thing that will never happen in MN is a referendum like you're seeing in the 8 states that have passed this in recent years. The MN Constitution doesn't allow for such "ballot questions", only for actual constitutional amendments. I strongly support legalization (and enjoyed it in trips to CO and WA last year), but most rational people would agree that it would be inappropriate as an amendment to our state constitution.
The bottom line is this will have to go through the legislature. There is no way for voters to directly petition the state. That's why it's really important to contact your state representatives on issues that are important to you, especially when there is pending legislation like there is right now on this issue.