beidseitig
on both sides
Located or taking place on both sides of something.
Bitte drucken Sie das Dokument beidseitig, um Papier zu sparen.
Please print the document on both sides to save paper.

on both sides / mutual
Loading your saved state.
on both sides
Located or taking place on both sides of something.
Bitte drucken Sie das Dokument beidseitig, um Papier zu sparen.
Please print the document on both sides to save paper.

The word is made up of 'beid-' (from 'beide', both) and '-seitig' (from 'Seite', side). So it literally means 'on both sides' or 'concerning both sides'.
Die Verhandlungen erforderten beidseitige Kompromisse.
The negotiations required mutual compromises.
'Beidseitig' (on both sides/mutual) can stand not only as an adjective before a noun, but also as an adverb describing a verb. In this function, it often comes at the end of the sentence.
Das Formular muss beidseitig ausgefüllt werden.
The form must be filled out on both sides.
You can recognize the meaning from the context. If it is about physical objects (paper, road), it means 'on both sides'. If it is about abstract things (contracts, feelings), it means 'mutual'.
Die Straße ist wegen Bauarbeiten nur einseitig befahrbar, nicht beidseitig.
Because of construction work, the road is only passable on one side, not on both sides.
In the meaning of 'mutual', 'beidseitig' is often used in formal or legal contexts. Typical combinations are 'beidseitiges Einvernehmen' (mutual agreement), 'beidseitiges Interesse' (mutual interest), or 'beidseitiger Respekt' (mutual respect).
Der Arbeitsvertrag wurde in beidseitigem Einvernehmen aufgelöst.
The employment contract was terminated by mutual agreement.
mutual
'gegenseitig' and 'beidseitig' can both mean that two sides are involved. 'gegenseitig' is used mainly for actions, feelings, or relationships: both sides do something for or toward each other. 'beidseitig' often sounds more factual and works well when something is located on both sides of something, or when two parties are affected by a rule, obligation, or agreement. Learners may confuse the words because 'beidseitig' can sometimes seem similar to 'gegenseitig' in contracts. In everyday language, however, people are more likely to say 'gegenseitige Hilfe' (mutual help), and more likely to say 'beidseitiger Druck' (pressure on both sides), 'beidseitige Schmerzen' (pain on both sides), or 'beidseitige Zustimmung' (mutual consent).

Der Vertrag kann nur mit beidseitiger Zustimmung geändert werden.
The contract can only be changed with mutual consent.
In unserer Familie ist gegenseitiges Vertrauen sehr wichtig.
In our family, mutual trust is very important.