Mindswap (as I was going to point out but James from NZ beat me to it) is the title of a novel by Robert Sheckley which matches your description very well. A shorter version by the same title, a novella, was originally published in Galaxy Magazine, June 1965, which is available for free at the Internet Archive. The ISFDB indicates that the novella has been translated into French ("Transfert stellaire") and Italian ("L'uovo di Ganzer"); there may very well also have been a Russian translation, although the ISFDB doesn't know about it. The excerpts below are from the Galaxy novella.
In the story people use some sort consciousness transfer technology to travel to distant solar systems.
In the final analysis, it was a matter of economics. Interstellar travel in the flesh was expensive, out of the question for an average sort of fellow. Unless, of course, he wished to avail himself of the advantages of Mindswap.
So the main character decides to travel and swaps bodies with an alien.
Yes, Marvin Flynn swaps bodies with a Martian:
In New York, Marvin went directly to the Body-Brokerage House of Otis, Blanders and Klent. He was sent to the office of Mr. Blanders, a tall, athletic man in the prime of life and already, at sixty-three, a full partner in the firm. He explained to this man his purpose in coming.
"Of course," Mr. Blanders said. "You have reference to our advertisement of Friday last. The Martian gentleman's name is Ze Kraggash, and he is very highly recommended by the rectors of East Skern University."
However, he/she isn't able to return back to his/her own body
Unfortunately for Marvin, Ze Kraggash is a crook, who has absconded with Marvin's body, leaving Marvin in a stolen Martian body which must be returned to its rightful owner:
The telepath handed Marvin a Dispossess Notice. Flynn accepted it sadly, yet with resignation. "I suppose," he said, "that I had better go back to my own body on Earth."
"That," the telepath said, "would be your wisest choice. Unfortunately, it is not possible at the moment."
"Not possible? Why not?"
"Because," the telepath said, "according to the Earth authorities, whose telepathic reply I have just received, your body, animated by the mind of Ze
Kraggash, is nowhere to be found. A preliminary investigation leads us to fear that Ze Kraggash has fled the planet, taking with him your body and Mr. Aigeler's money."
Marvin hires a detective:
"Aren’t you Madame Ripper de Lowe, the transvestite who was robbed last night in the Red Sands Hotel?"
"Certainly not. I'm Marvin Flynn. I lost my body."
"Of course, of course," Detective Urdorf said, nodded vigorously. "Let's take it point by point. Do you remember by any chance where you were when you
first noticed that your body was missing? Could any of your friends have taken it as a joke? Or could you have merely misplaced it, or perhaps sent it on a vacation?"
and instead is transferred to another alien's body and so on.
The next thing he knew, he was in a Melden body, on Melde.
As the story progresses the protagonist jumps from body to body until he/she slowly loses his/her own identity.
With a long rolling stride and a creaking of leather boots, Marvin Flynn strode down the wooden sidewalk. Faintly there came to him the mingled odors
of sagebrush and chapparal. On either side of him the adobe walls of the town glittered under the moon like dull Mexican silver. From a nearby saloon there came the strident tones of a banjo —
Frowning deeply, Marvin stopped in midstride. Sagebrush? Saloons? What was going on around here?
