Yes, they do have some special rights, mostly having to do with employment issues. In Minnesota, members of the Legislature are "citizen legislators" and many have jobs outside the Legislature. For example, a member of the Legislature who is employed in the private sector must be allowed to resume his or her old job or a position of similar seniority, pay, and status if he or she reapplies within 30 days after the end of the session. A member cannot be discharged because of time spent in legislative service, nor can they be denied their seniority or benefits.
In addition, no employer can discharge a member of the Legislature in retaliation for statements made or beliefs held in his or her capacity as a legislator.
If the legislator is employed by a public entity, such as a city or a school district, he or she must be restored to his or her original position or a position of similar status. In addition, that person is entitled to an unpaid leave of absence during any or all of his or her term of office.
The state constitution provides that members cannot be arrested while the Legislature is in session or when they are on their way to or from session, except in cases of felony, treason, or breach of the peace (see Sec. 10. Privilege from arrest). This privilege applies to misdemeanors or gross misdemeanors, excluding what the courts consider a breach of the peace. (Examples: assault or threatened assault, breaking and entering, driving while intoxicated, speeding, violent verbal attacks, or other acts that cause serious harm to people in the vicinity.)
The privilege only prevents detaining a member during session; it does not provide immunity from prosecution after session ends. The protection is not automatic. A member can either assert the privilege in court or choose to waive it.