You can choose to do 2 relationships, a many-to-many and a one-to-many.
Or use a single relationship a many-to-many and flag the type of the relationship in the pivot table.

Personally I would go with two relationships if there could be a single owner of the Project and type of members in a Project doesn't have high importance.

The project will only have one owner and members will not have different levels of importance (sticking to basics at the moment).

If I understand correctly would you suggest that User and Project have a many to many relationship and Project has a belongs to relationship with User?

Something like...

<?php
class User extends Eloquent
{
    public function ownedProjects()
    {
        return $this->hasMany('Project', 'owner');
    }

    public function joinedProjects()
    {
        return $this->belongsToMany('Project');
    }
}

<?php
class Project extends Eloquent
{
    public function owner()
    {
        return $this->belongsTo('User', 'owner');
    }

    public function members()
    {
        return $this->belongsToMany('User');
    }
}