When is a poem finished? The decision on whether your poem is complete depends on who is involved. By “complete”, I do not mean perfect, but stable such that no further changes may be foreseen by those persons it is reviewed by. These persons may be the author, trusted readers, workshop participants, editors, or the readership.
With a distribution of one, just the writer, can you conclude that it is complete? Or is this just a rough sketch, like a bozzetto of a sculpture or a renaissance painting. Do you feel that there is completion in the sense of the poem being “locked-in”, where every word, every phrase has been examined? And no further changes can be made? All editing has been done. All possible interpretations by readers have been considered?
I remember some advice given during a creative writing course that when you think you have finished a short story, you should go back and edit out another third of the text! This always felt impossible, until it was done and the result to me usually felt much improved. With a sonnet you will be subject to this sort of pressure anyway since you are limited to fourteen lines.
However, when your work is shared with a trusted reader, there may be further suggestions made requiring edits or even rewrites and then eventually an arrival at a new sense of completion.
When an author goes back to a draft poem, left unread for a couple of weeks, they may also have further insights leading to changes. They can see it with fresh eyes. You should try this.
When shared with a wider group, such as a workshop group, even more comments may be forthcoming.
Finally, one could say when your work has been published, subjected to editorial review, then it is for all intents and purposes finished. Even though some well-known poets have made minor adjustments to subsequent versions appearing in anthologies.
Conversely, take care not to overwork a poem, where word choice become too ornate or precious, where you change words just to fit rhymes but that don’t really fit the text and that look too obvious. If there is too much exposure to workshops and poems can become over-workshopped. As always, there is a balance to be sought.
Some poems have that X-factor and stand out like beacons. This is difficult to prepare for, you must write and write and get better all the time using all the feedback possible. I believe, creativity is like a muscle that can be trained to some extent. Poets should critically read other poets work, e.g. asking what works well and why? The more you read and write and flex this muscle the more chance you will have of creating one of these beacons of poetry.
When you get there, and the poem is finished, the satisfaction is immense.