Let's say there is a Jewish community where everybody knows each other, for example in a city. Let's say a new Jewish person arrives and manifests their will to get integrated with them. Nobody knows this person there. On what basis would that person be taken as someone belonging to the community? Stretching the scenario, how likely is it to impersonate a Jewish person and deceive others? If it is a man, wearing a yarmulke might serve as a visible, external, sign; but surely that is not enough. Is a newcomer examined somehow? What is checked in these cases? Are there "Jewish identity cards", or documents? That could show his affiliation to a synagogue or institution that could help verify one individual's claims, or a certificate of conversion if it is the case. If there is such document, how rude is it to ask that to a newcomer?
There’s no Jewish identity card! In theory, people who claim to be Jewish should generally be believed absent any specific reason to doubt them. For example, Rambam said (Hilchot Issurei Bi’ah 13:10) that somebody who claims to have converted to Judaism elsewhere is to be believed. In practice, though, standards vary. Some synagogues might demand varying levels of proof, whether a letter from a previous rabbi or a copy of the individual’s parents’ marriage certificate. Here is a detailed, though perhaps slightly dated, review of the Jewish law on this topic and its application in Reform synagogues.