"I see Martha with her friend."

Translation: Widzę Martę z jej przyjacielem.

[polec2]

If we remove jej, will it mean the same thing or is this only possible with my friend ("Jestem z przyjacielem" instead of "Jestem z moim przyjacielelm) ?

[Jellei]

It will be okay and most likely still mean "her friend". Although it could be a bit ambiguous then. Anyway, added an option without 'jej'.

[plastoquinone]

Is the friend here Martha's own friend, or the friend of another woman? Thank you.

[immery]

technically could be both, but it's more probable it is Martha's friend. (This sentence works just like the English one, using "swoim" here, would mean "I see Marta with my friend")

[Xelialith]

Can't you write: "Widzę Martę ze swoją przyjaciółką"?

[Jellei]

That sentence makes sense, but it is not a translation of the English one. After all, "I" am the subject of the sentence. As "swoją" also applies to the subject, that means that the friend is mine, not Marta's.

[gitaarfreak]

Can friend in this context mean boyfriend?

[Okcydent]

Unlikely. In Polish boyfriend is "chłopak" or eventually "partner" or "facet". But with the latter we should use some possessive pronouns ("jej facet") to indicate that that man is somewhat special to her.
Generally, with term „przyjaciel” all the things related to "friendzone" come into play.