寂しい and 悲しい: loneliness, absence, and sadness
Compare loneliness or desolation from absence with sadness about an event, including farewells where both feelings can overlap.
Feel someone's absence
友達が引っ越して、寂しくなりました。 says that someone became lonely after a friend moved away. 寂しい, read さびしい, fits the sense of missing someone's presence.
誰もいない駅は寂しい感じがしました。 says that the empty station felt desolate. 寂しい can describe a place's impression as well as a person's feeling.
Respond to something sad
悲しい知らせを聞いて、涙が出ました。 says that someone heard sad news and tears came. 悲しい, read かなしい, expresses sadness about loss or something distressing.
この映画の結末は悲しかったです。 says that the film's ending was sad. The feeling can concern the story regardless of whether the viewer watched alone.
Feelings that can coexist
別れるのは悲しいし、一人になるのも寂しいです。 says that parting is sad and being alone is lonely. The same event can involve both sadness at separation and loneliness afterward.
一人でも寂しくありません。 means that the speaker is not lonely even when alone. Being alone does not guarantee loneliness.
Does 一人でも寂しくありません say that being alone is necessarily sad?
Show the answer and why
No. It says that the speaker is not lonely even alone. It negates loneliness without separately stating whether there is sadness.