Corporate jet Captain reported another aircraft pilot performed a go-around due to the close proximity of Captain's aircraft during climb-out. Captain suggested the airport needs a temporary Tower or VFR traffic restriction due to the increase of flight traffic during the winter.

Narrative

We were preparing for takeoff on Runway 10 at HDN. Conditions were VFR. However; we always operate on a IFR flight plan. So Denver Center released us for departure with a 3-minute void time and referenced a VFR Aircraft Y on a 5-mile final. We then switched to CTAF and made a radio call. However; it is possible my copilot did not do this correctly or inadvertently did not depress the button fully to switch to the other radio. The pilot of Aircraft Y said he did not hear us. We then visually scanned final and did not see any traffic; and proceeded with our takeoff without incident. However; when switching back to Denver Center on the climb-out we heard the pilot of Aircraft Y report that he had to do a go-around because he was too close to us. I am unsure just how close he was as we never saw him either before or after the takeoff. I believe he initiated the go-around about a mile out but I am unsure. He then reentered the pattern and landed without incident.I believe safety at the HDN Airport needs to be reviewed. It is an extremely busy airport in the winter with a heavy load of both airlines and corporate aircraft; yet it is uncontrolled. There have been several incidents in the past very similar to ours. Most notably; about a year ago; an airliner had a tail strike because they had to rotate early to avoid another aircraft that was landing the opposite direction. I believe a temporary Tower in the winter or restricting VFR traffic in the winter would help tremendously. It is only a matter of time until a major incident happens at this airport if it continues to operate uncontrolled.