For field linguists, this isn't unheard of, but in most cases there is a contact language or lingua franca in the region which allows more extensive communication. The University of Toronto maintains a bibliography of fieldwork resources which you may find interesting. Most of my knowledge of field methods in early documentation work comes from Robbins Burling's book Learning a field language and my discussions with Kate Lindsey, a field worker who documented Ende. She describes a bit of her initial work in this news article.
In field situations, linguists will often use elicitation sessions to gather data. In these settings they will often try to elicit basic vocabulary to develop a preliminary lexicon from which they can make other generalizations. These elicitation session can use a number of methodologies. Linguists can use images or flashcards to learn what words associate with what objects. Linguists can use actual objects in a similar manner.
More complex grammatical patterns can be elicited through methods like map tasks. In these tasks, speakers are paired up and both are given copies of a map. The instruction giver has a map with a route they need to get the instruction follower to draw on their own blank map. Because the route is known to the linguist, the correspondence between phrases and meanings can be deduced. If the word "gavagai" occurs every time the speaker wants the hearer to draw a line to the left, the linguist can make an informed guess that "gavagai" means left. As the number of examples grows, more complex grammatical structures can be uncovered through comparison. If you have examples such as "he sits" "he sat" and "she sits", a linguist can determine what sound sequences correspond to what meanings by comparing what meanings share sound sequences across examples. From these the linguist can develop hypotheses about new sentences they have not seen before.
As the linguist builds up understanding from these tasks, they can begin to ask questions of their collaborators and do quasi-experiments with their grammar. They may quickly learn how to ask questions, and so it is possible the linguist can ask their collaborator "Is [...] a good sentence?" As the linguist learns to speak the language, they will also try to use it. Through using the language, they get positive and negative feedback about whether the grammar they deduced is correct. They may also get feedback from their collaborators on mispronunciations, or better ways to phrase things.
Once you get past the initial hurdles of establishing basic communication and collecting a basic corpus, further work becomes more similar to any other field situation. More complex patterns can be gleaned through analysis of the corpus, and those methodologies are similar to solving a phonology or morphology problem. More complicated tasks can be used to gather new data, and hypotheses can be tested through acceptability judgments.
