SF340 First Officer reported an engine failure during climb out. Flight crew returned to departure airport and landing resulted in all tires going flat.

Narrative

On Day 0; XA15; I experienced an engine failure of the right engine in flight that resulted in an immediate landing. I was serving as SIC and Pilot Monitoring on a Saab 340B; operated by Company on a revenue cargo run. With me was Person A; who served as PIC and Pilot Flying. The conditions at ZZZ were VRB04KT XXSM FEW0XX FEW1XX FEW2XX 37/13 A30XX. The density altitude was around 7500 ft; and the runways in use were Runway XX and XYL. After loading cargo and fuel; we calculated our gross weight to be at 26;160lbs; our CG was in a safe range at 22%. We used the App for takeoff calculations; and had determined the need for a reduced power take off due to the higher density altitude and high temperature. We elected to use Runway XX due to it being 3000 feet longer; and did not have a mountain 4.5 miles beyond its departure end like Runway XYL had. On engine start up; there was no indications of abnormalities on the right engine. We continued with our checklists; then requested taxi to the runway. Thereafter; we taxied to Runway XX; and still noticed no abnormalities. Tower gave us clearance to back taxi down Runway XX then when ready fly left turn heading 120; climb to 17000; and clear for takeoff. After lining up and performing pre-takeoff checklists; we still had no indication of an abnormal situation. All engine instruments were within the green arc; and we were on the go for takeoff. During the roll I pointed out that our right engine was getting hot; but still within the green arc. After rotation; everything was going normally. Tower handed us off to Departure and we continued climbing.I checked in with Departure; and at approximately 1000 ft AGL; Person A called for the climb power/climb checklist. As I was setting the power; the right engine made a loud bang; followed by a 'Right Ignition' light illuminating; then settled. A few seconds later; the right engine made multiple loud banging noises. At this moment we recognized this was a compressor stall and began to run our memory items. However; we were unable to recover the engine; and it lost all power and now had a windmilling propeller. I [requested priority handling] with Departure Control; and began running through our quick reference handbook for a failed engine.Departure gave us clearance to land on any runway; and we selected Runway XYL. I was able to complete the QRH; resulting in a feathered and secured right engine. Additionally; we performed pre-landing checklists; and maintained a safe airspeed. We completed a safe landing on Runway XYL; and exited at Taxiway 1 with no issues. The ground ops truck met us on Taxiway 2; and followed us to the FBO Ramp. Upon turning right onto Taxiway 3 the ops truck said on frequency one of our main tires appeared flat. A few seconds later; we began to feel the aircraft shake while rolling. We elected to stop just past the movement/non-movement line on [Taxiway] 3; and requested the truck verify if one or all tires were flat. The truck confirmed all tires were flat. So; we requested shutdown on Taxiway 3.We then secured the airplane; and exited the aircraft on [Taxiway] 3. The tires did not appear as if they had skid on the runway; and the brakes had smoke coming off them. Eventually the aircraft was towed off of 3 and onto the FBO ramp.