beiderseits
on both sides
On both sides of something. Used as an adverb or as a preposition with the genitive.
Beiderseits der Straße standen hohe Bäume.
On both sides of the road stood tall trees.

on both sides / mutual
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on both sides
On both sides of something. Used as an adverb or as a preposition with the genitive.
Beiderseits der Straße standen hohe Bäume.
On both sides of the road stood tall trees.

The word is made up of 'beide' (both) and the ending '-seits', which means 'on the side.' Literally, it therefore means 'on both sides.'
Beiderseits des Weges blühten Blumen.
Flowers were blooming on both sides of the path.
'Beiderseits' (on both sides / mutual) belongs more to written language or to a more formal style. It often sounds somewhat elevated.
Die Vorteile der Kooperation liegen beiderseits.
The advantages of the cooperation exist on both sides.
In the spatial meaning, 'beiderseits' (on both sides) is often used as a preposition that requires the genitive. However, it can also stand alone as an adverb.
Beiderseits des Rheins gibt es viele Burgen.
There are many castles on both sides of the Rhine.
You can recognize the mutual meaning from abstract nouns that express a relationship or agreement. Typical words are 'Interesse' (interest), 'Einverständnis' (consent), or 'Vorteil' (advantage).
Der Respekt war beiderseits groß.
The respect was great on both sides.
mutual
'gegenseitig' and 'beiderseits' can both mean that something comes from two sides. 'gegenseitig' is used very often with people or groups who do or feel something for one another. 'beiderseits' sounds more formal and can also be meant spatially, that is, 'on both sides' of a road, border, or river. Learners may confuse the words because 'beiderseits' sometimes means something similar to 'gegenseitig'. In everyday language, 'gegenseitig' is usually more natural for help, trust, or respect.

Beiderseits der Straße stehen alte Bäume.
There are old trees on both sides of the road.
Die Schüler helfen sich gegenseitig bei dem Projekt.
The students help one another with the project.