gemeinsam
together
As an adverb, it describes an action that is carried out together with other people.
Wir sind gemeinsam in den Urlaub gefahren.
We went on vacation together.

together / shared
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together
As an adverb, it describes an action that is carried out together with other people.
Wir sind gemeinsam in den Urlaub gefahren.
We went on vacation together.

The word 'gemeinsam' (together/shared) is closely related to the noun 'die Gemeinschaft' (the community). Both words share the idea that something connects a group of people or is shared by them.
Die Dorfgemeinschaft organisierte ein gemeinsames Fest.
The village community organized a shared festival.
'Gemeinsam' (together/shared) can stand as an adverb without an ending or be declined as an adjective before a noun. As an adverb, it describes how something is done. As an adjective, it describes what kind of thing something is.
Wir kochen gemeinsam. (Adverb)
We cook together. (Adverb)
When 'gemeinsam' (together) is used as an adverb, it often appears with verbs that describe a social activity. Typical combinations are: gemeinsam etwas unternehmen (to do something together), gemeinsam entscheiden (to decide together), gemeinsam arbeiten (to work together), or gemeinsam lachen (to laugh together).
Lasst uns am Wochenende gemeinsam etwas unternehmen.
Let's do something together this weekend.
As an adjective, 'gemeinsam' (shared/common) often describes abstract things that are shared. Common combinations are: gemeinsame Interessen (shared interests), ein gemeinsames Ziel (a shared goal), eine gemeinsame Zukunft (a shared future), or gemeinsame Freunde (mutual friends).
Trotz unserer Unterschiede haben wir ein gemeinsames Ziel.
Despite our differences, we have a shared goal.
with
'mit' and 'gemeinsam' can both show that several people are involved. 'mit' usually names the person or group that is there too: 'mit meiner Freundin' (with my friend). 'gemeinsam' puts more emphasis on doing something together, or on something belonging to or concerning several people. It can be used as an adverb or as an adjective, for example 'gemeinsam arbeiten' (to work together) or 'ein gemeinsames Ziel' (a shared goal). The words can be confused because they often occur in similar situations. 'mit' connects an action with a person; 'gemeinsam' describes the action or thing itself as shared or done together.

Mein Freund und ich haben ein gemeinsames Konto für die Miete.
My boyfriend and I have a shared account for the rent.
Du kannst die Bewerbung mit deiner Freundin noch einmal lesen.
You can read through the application again with your friend.