A DM may allow a player to train, learning a language or tool proficiency in their downtime. In order to train, a player needs the following:

• Instructor: An instructor is needed to learn a language or a tool. Xanathar clearly states "given enough free time and the services of an instructor...". As such, a player in the middle of an empty forest could not learn a tool or language. However, the player could learn a tool or language from another character in the party (of course, the "teacher" would need to spend their downtime teaching).
• Time: Training takes about 10 workweeks, minus a number of workweeks equal to the character's Intelligence modifier (an Intelligence penalty does NOT increase the time needed).
• Money: Training costs 25 gp per workweek.

Training Complications

Complications while training typically involve the teacher. There is a 10 percent chance for every 10 workweeks spent training that a complication occurs. A DM can choose whether to use complications, and can create their own complications if they wish.

d6	Complication
1	Complication:  Your instructor disappears, forcing you to spend one workweek finding a new one.†
2	Complication:  Your teacher instructs you in rare, archaic methods, which draw comments from others.
3	Complication:  Your teacher is a spy sent to learn your plans.†
4	Complication:  Your teacher is a wanted criminal.
5	Complication:  Your teacher is a cruel taskmaster.
6	Complication:  Your teacher asks for help dealing with a threat.

† Might involve a rival.