Multi-Stage Aggregation SOlution

db.getCollection('versus-challenges').aggregate([{
    $match: {
        type: {
            $in: ['public','private']
        }
    }
}, { // "uuids" is an array of uuids, so we split it
    $unwind: '$uuids'
}, { // Index by uuid+type, and add the challenges ID in an array
    $group: {
        _id: {
            uuid: '$uuids',
            type: '$type'
        },
        challenges: {
            $addToSet: '$id'
        }
    }
}, { // Split the array of challenge ID into multiple document to be able to $group again on them
    $unwind: '$challenges'
},{ // Group by uuid & type, count how many of each
    $group: {
        _id: {
            uuid: '$_id.uuid',
            type: '$_id.type'
        },
        challenges: {
            $sum: 1
        }
    }
},{ // Re-assemble into a single document per uuid
    $group: {
        _id: '$_id.uuid',
        types_played: {
            $sum: 1
        },
        played_total: {
            $sum: '$challenges'
        },
        stats: {
            $push: {
                type: '$_id.type',
                played: '$challenges'
            }
        }
    }
}, { // Sort to get the users who played the most on top
    $sort: {
        played_total: -1
    }
}])