Not necessarily... I'm speaking for the US now. Every airport with an instrument approach is assessed for an IFR departure. If you want to know the details, you can read about it in the FAA's Instrument Procedures Handbook: Instrument Procedures Handbook (IPH) | Federal Aviation Administration (faa.gov). If there are no obstacles or terrain affecting an IFR departure from the runway, the pilot is free to turn in any direction after climbing to 400' above the runway elevation and proceed on their way. It's important to note that this "diverse" departure assessment is valid only out to 25 miles in nonmountainous areas and 46 miles in mountainous areas. If obstacles or terrain effect the IFR departure, then takeoff minimums and/or an obstacle departure procedure (ODP) is published for that runway. The ODP can be nothing more than listing of low, close in obstacles that must be seen or otherwise avoided on takeoff. These are not a factor in turbojet takeoffs as long as all engines are operating. The takeoff performance requirements mean that you are well above most of these obstacles by the time you cross the end of the runway. The ODP can consist of a higher than standard climb gradient that will clear obstacles or higher than standard takeoff minimums that allow the pilot to see an avoid the obstacles. The ODP can also turn restrictions, such climb on a heading to an altitude before turning in a specific direction. Finally, the ODP can consist of specific routing to avoid obstacles, to include a climb in holding until reaching a safety altitude.
A SID is nothing more than an abbreviated ATC clearance in charted form. The route is evaluated for obstacle clearance, and if assigned by ATC you fly it instead of the ODP. If no SID is assigned, but takeoff minimums and/or an ODP is published for the runway, and you are an FAA certificated operator (i.e., airline or charter operator), then you're required to fly the ODP.
If there is no SID or ODP, then you are free to turn in any direction and proceed on your way to your first en route fix. However, again you are responsible for terrain and obstacle clearance once you're more than 25/46 miles from the airport.
Hope this helps,
Rich Boll