The US Navy investigates their missing ships themselves.
This might belong to history, but lost ships have real world examples. For a missing navy ship, there's a perfect example: USS Cyclops was US Navy property and went missing in March 1918. On her last journey, she was 25% overloaded with manganese, drove on one of two engines due to cracked cylinders, and performed a (possibly unscheduled) stop in Barbados. Allegedly, she was deep in the water and on a previous journey, she'd bend under waves. She left Barbados on March 4 and was expected in a US port around March 13.
For easy reference, Chesapeake Bay is about 1700 sea miles (3500 km) away from Barbados. Her listed cruise speed using both engines was 15 knots. At a mediocre pace of 10 knots, she'd have made the trip in about 6 to 7 days, and to arrive on March 13, she'd be traveling on average with a lazy 7.8 knots. However, no message arrived and she stayed overdue.
Contemporary shipbound radio equipment had an effective broadcast range of somewhere around 150 nautical miles (+-50) on an average day, and her expected course would lead her to traverse the ocean far beyond that reach for the majority of the travel. The Titanic was heard at least 150-180 nautical miles in 1912 because she had exceptionally calm weather, there was little air traffic at night, and her much longer antenna was mounted very high. Under ideal conditions, 250-300 nautical miles range was the most that could be acieved.
On April 15, the Navy themselves did give a statement that they were expecting the ship lost but continued the search, on June 1st the ship was finally declared lost at sea.
The 1918 Annual Report of the US Navy reads on page 28:
Loss of the Collier "Cyclops"
There has been no more baffling mystery in the annals of the Navy than the disappearance last March of the U. S. S. Cyclops, Navy collier of 19,000 tons displacement, with all on board. Loaded with a cargo of manganese, with 57 passengers, 20 officers, and a crew of 213 aboard, the collier was due in port on March 13. On March 4 the Cyclops reported at Barbados, British West Indies, where she put in for bunker coal. Since her departure from that port there has not been a trace of the vessel, and long-continued and vigilant search of the entire region proved utterly futile, not a vestige of wreckage having been discovered. No reasonable explanation of the strange disappearance can be given. It is known that one of her two engines was damaged and that she was proceeding at reduced speed, but even if the other engine had become disabled it would have not had any effect on her ability to communicate by radio. Many theories have been advanced, but none that seems to account satisfactorily for the ship's complete vanishment. After months of search and waiting the Cyclops was finally given up as lost and her name stricken from the registry.
Who were the ships that searched? The Washington Times tells, in a retrospective, that it was US Navy Cruisers. The secretary of the Navy, Daniels, was ultimately responsible for the search - and did the relevant statements. And the Annual report above also indicates so. As of 2023, the ship is still lost to the sea.
Her sister ships USS Proteus AC-9 and USS Nereus AC-10 were sold to Canada in 1941 and lost at sea in November and December 1941 for unknown reasons (including possibly U-Boats, though no claim exists) and without a trace in the Caribbean while hauling Bauxite. Their last sister, USS Jupiter AC-3 became USS Langley CV-1 in 1920, was downgraded to AV-3 in 1937, and was scuttled after a torpedo hit in 1941. In her upgrade, most of her facilities were rebuilt.
Insured private ships have insurers pay investigators
Only private ships that are insured (which in the 1920s were by far not all) will have the insurance company try to identify a cause of the loss. One such ship insurer agency that was active in the time is Lloyds of London.
Besides them, there are also classification societies like Germanischer Lloyd or Lloyd' Register that handle insurer, underwriter and shipowner relations by publishing lists of ships and their condition. Often enough, they would act as intermediaries for the owners/insurers, technical advisors to shipyards, and inspectors. Germanischer Lloyd cooperated with the sailor's union and by that even managed to enforce safety standards. On the grand scale, the classification agencies indirectly dictate insurance rates because they inspect vessels to be insured for seaworthiness, and a bad rating results in bad rates. At least in the early 1900s, some of them also assisted both parties as a neutral agency to manage cases of ship loss.
Generalities of a claim handling
If the ship was presumed lost, the shipping company would notify the relevant company.
The insurer or intermediary would then start with an assessment: weather reports, the latest reports about the ship's maintenance, route, and cargo manifest as per the last harbor would be compiled.
Depending on the amount of the policy and resources, investigators would be sent to the last ports to try and identify problems with the ship, and crew and to augment the other reports already found and presented.
Their report would then join the stack of papers for the claim assessor at the insurance. Classification agencies like the Germanischer Lloyd often added their own paperwork (because they had inspected the ships somewhat regularly) and mediated a possible settlement between the shipowner and insurer if needed.
In general, the certification agency's estimation about seaworthiness of a vessel might preclude a deeper inspection: A ship deemed to be a "Seelenverkäufer" (~coffin ship / not seaworthy) would have the claim massively reduced.
For how one claim might be handled in legal circles, including the ensuing lawsuits against relevant people, the Lusitania and Titanic might serve as very public examples.
Who searches for the missing ships?
Now, this gets interesting: We are knee-deep in the Laws [and customs] of the Sea. Hearing a distress call would, under the 1920s require ships to relay such and then assist in a search and rescue if possible, and then report in port when and where they heard the call.
For a somewhat working example of some parts of this, again, the 1912 sinking of the Titanic is a good example: The SOS CQD signal was answered by two ships in the closer area: RMS Carpathia risked man and mouse to try and get to the site of the sinking at 17 knots top speed over 58 nautical miles, arriving still in the dark. SS Mount Temple also answered but was delayed by pack ice. She'd join the search in the early morning.
The nearby SS Californian - the very ship that had informed Titanic about icebergs - had stopped for the night to not risk a disaster, but also was without a radio operator during the night. While the acting officer could see the rocket signals, he and the captain could not make sense of signals. She would turn as soon as the emergency became apparent to her by the radio operator between 4 and 5 AM, and she would be joining Mount Temple in the search. She recieved the information from the inbound SS Frankfurt.
After picking up survivors before the break of dawn, Carpathia aborted her transatlantic cruise and returned to New York to handle the injured, the other two vessels searched the area for some hours fruitlessly before resuming their voyage. SS Birma had answered the Titanic's call from 100 nautical miles away within short time of it being sent and set course to the location, arriving after the rescue was concluded. Her offer of supplies to the Carpathia were denied.
While SS Frankfurt was about 150 nautical miles from the accident, she was actually the first to anwer the call at all, close to midnight. She had a comparatively powerful radio and steamed through the night preparing for survivors around 1 AM. She was expected to arrive at the scene around 11 AM. She also was responsible for relaying the message what happened to the Californian.