
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.

mutual
Coming from both parties involved; mutual.
Das Einverständnis zur Vertragsauflösung war beiderseits vorhanden.
There was mutual consent to terminate the contract.
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 express that two parties are involved in an action. Gegenseitig emphasizes that the action is directed toward each other, while beiderseits initially indicates only that it originates from both sides. Use beiderseits for contributions or statements from both sides and gegenseitig for a direct action done for each other.

Beiderseits wurden Vorschläge gemacht, um die Verzögerung zu beenden.
Both sides made suggestions to end the delay.
Die beiden Parteien machten sich gegenseitig Vorwürfe wegen der Verzögerung.
The two parties blamed each other for the delay.