Flight crew reports departing BZN from Runway 12 on runway heading and is informed by ATC that they should have flown the published procedure even though one was not assigned.

Narrative

Departing BZN our clearance was as filed; [maintain] 37;000. Our routing on the release was N0452F370 DCT BZN V86 BIL... The ceilings were around 10;000 - 10;500 FT. On departing Runway 12 we flew runway heading and were handed over to departure. Pilot not flying called Departure and reported altitude and runway heading. Departure asked if we had been assigned runway heading and we replied we believe so; nothing was assigned on the departure just cleared for take off. Departure then advised us to turn to our next fix which was V86 airway after BZN. We then got into a conversation about what departure procedure we had used and we told ATC we had not been assigned a departure procedure. It was our understanding you fly runway heading unless assigned a procedure. Departure replied yes normally but out here in Montana we do things a little different. He said how you depart is PIC discretion; and that the Minimum Sector Altitude in the area was 11;500. So we should have used a departure procedure. We said we should have been assigned a departure procedure. We both thought a departure procedure had to be assigned. We did not realize you could just decide to do one or not; or even decide to do one without being cleared to do so. Departure said that this exact thing happened the week prior and that something needed to be addressed. He then sent us over to Minneapolis Center.Dispatch should assign a departure on the release if weather is below 11;500. Put out a memo to pilots that BZN 12/30 requires departure procedure if ceilings below 11;500 AGL. Tower should have referenced BOZMAN 3 departure on takeoff clearance or Departure should have on the switch over.