"Let's sit together."
て form is a connecting form. It is not a request form. If you hear sentence ending with て, it is not a grammatically correct sentence. It is often the verbal abbreviation of て ください。
Let Us is a suggestion and in Japanese it is expressed by the indefinite verb form ましょう. It is close to the English conditional form I think. The sentence can be understood literally as "Would we sit together?"
Oh okay that makes a great deal of sense. I saw people in comments earlier saying that you should use て form when making requests, such as ください. I assume then, that is generally right but not accurate.
On a technical level "let us" is a request but you are perfectly correct that the English "let's" is used as a suggestion.
Can you just say, "すわりましょう" and omit "いっしょに"?
I've tried looking this up before as "ましょう" seems to indicate a proposal, so from an English perspective, it seems like "いっしょに" would be implied. This comes up when comparing statements like, "Let's sit" vs "Let's sit together" or "Let's go to the pool" vs "Let's go to the pool together", where the sentences can be used almost interchangeably.
I see that Marknld suggested something similar, but got downvotes; so I wanted to know if there is a more pronounced / different meaning in Japanese if "いっしょに" is omitted.