How can I locate a missing person abroad?

Will airlines, trains, hotels, etc. confirm whether he boarded or checked in?

Airlines and trains, probably not. Unless you have the PNR, and you can then probably log into the “manage by booking” section of the relevant sites which may give you some info (this would normally be an invasion of privacy, but if you have the PNR that means they are trusting you).

For hotels, just call the hotel and ask to talk to the person. If they have checked in they will transfer your call to their room. Otherwise they will most certainly tell you they didn’t.

In the worst case you can ask to leave a message (requesting a call back for instance) and they will pass it on.

If said person is traveling by public transport or good rental car, I would not worry if there is no contact for a couple of days. Only if the person in walking or cycling and has not reported in at the time they promised to do that I would try to search for him. And in that case I would start by calling his relatives to ask if they have been in contact.

Starting an international search through a foreign government (to the country where the person might be missing) is way out of balance for someone who has just not contacted a friend through internet.

It has happened to me, and others I know, loosing your phone or having it stolen, crossing a border and finding that your phone does not work for whatever reason in the new country. Just being so busy with life as a tourist that you forget to log on with your usual groups, and so on.

You may not be told whether he has used the hotels or transport he was supposed to be using. But if you call the hotel he is supposed to be ‘tonight’ or ‘tomorrow night’ and ask them to hand him a message to contact someone specific it should work.

In the past it was normal to travel around for months and only send post cards home and nobody worried if a card arrived a day or two late. Now we are connected almost 24/7 and start worrying as soon as contact is broken.
Somewhere in the middle is the right level of worry. For me it was calling home every two or three days, so you would get updated on news from both ends. And it was not uncommon to warn that the next call might be late if you headed out into the unknown.

Only if there is a good reason to find the person, like a seriously ill close relative they should see before passing away, I feel it is right to start a big search. And then first in the location where they are supposed to be, (and them not being in the hotel you expect them to be.)
The local police to that place might be the most efficient.

If we are able to get his booking details, will airlines, trains,
hotels, etc. confirm whether he boarded or checked in?

Not easily. Typically, airlines and hotels do not give out this type of information for privacy protection. However, you can always call and try: smaller hotels or bed & breakfast tend to be less stringent with this.

Some airlines and travel agencies allows a traveler to register an emergency contact. They may be willing to disclose basic information (flight taken or not taken, room checked in or not) to the registered person.

Otherwise you’d have to go through the authorities per Ozzy’s answer.

Ideally the authorities should be contacted when a person is believed to be missing. Now whether you want to do that now or later that’s up to you and the family, but I would definitely check with them first.

Once you make the decision to alert authorities, I would first alert the US Dept. of State:

U.S. Citizens Missing Abroad

If you are concerned about a U.S. citizen relative or friend who is traveling or living abroad, you may contact the nearest U.S. Embassy or Consulate, or call our Overseas Citizens Services office in Washington, D.C. at 1-888-407-4747 (from outside the United States and Canada, call 202-501-4444)

They will then check with the local authorities:

Our embassies and consulates abroad can use the information you provide to try to locate the individual and pass on your message. We can also check with local authorities in the foreign country to see if there are any reports of a U.S. citizen hospitalized, arrested, or otherwise unable to communicate with those looking for them. The more information you provide about the individual, the better we may be able to help.