In the last couple of weeks, I've been asked to talk a lot about point of view in poems. In workshops I often question whether the poem has enough risk or urgency. There are many ways to heighten tension in a poem. Some are thematic, e.g. alluding to backstory. Some are syntactical, e.g. phrasing a sentence as a question. Some are formal, e.g. breaking lines at a critical junction, or enjambment between stanzas.
But I think point of view is undervalued as a determinant of tension. The POV you choose helps shape the risks your poem can take.
First Person: Here, the central risk is one of discovery. The speaker's understanding of something, or the reader's understanding of the speaker, should change across the course of the poem. That doesn't mean the subject might not also do something. But keep in mind that you've chosen a POV that privileges his or her perception of that act/experience, a version that may or may not be reliable, versus focusing on the act/experience itself.
Second Person: Here, the central risk in one of disclosure between parties. A secret is being revealed or created by those present in the world of the poem. If the "you" is being addressed through a series of imperative commands, then he or she should be asked to do something counterintuitive to what we know of that identity.
There are a ton of Second Person poems being written right now, in part because it is a shortcut to intimacy with the reader. But it's frustratingly static when "I" tells "you" a story, across the course of the poem, that in reality would already be known and complete between the two parties. It's a gimmick, much like when the character in a short story pauses on a doorstep and flashes back to an entire romance right while her finger is pressing the theoretical buzzer.
Third Person: Here, the central risk is dramatic. These are characters, and you control their stage, even if your writing is inspired by contemporary or historic events. A compelling Third Person poem, whether bird's-eye (in which you're battling the drag of expository language) or omniscient (in which you're tackling the beast of authenticity), is an awe-inspiring thing; I wish more people would try their hand at them.
Ask yourself why your draft uses its particular point of view. Try envisioning the same poem in First Person, Second, Third. What does an outside view reveal or emphasize about your "characters" and their dynamics? What secrets would one tell the other? How do you newly sympathize (or not) when an antagonist becomes the speaker?
When the poem finds its destined POV, it will cling to it. Your favorite moments won't work in the other modes. You can try the same thing with verb tense: rotate the poem through past, present, and future. And I always create an intermediate draft in which all line and stanza breaks are erased. I massage the syntax as a prose-paragraph, then I break again. Sometimes this results in the same visual format. Sometimes not.
When the poem starts to fight back, to commit over and over to certain aesthetics, that's when I know I'm on my way. And I'm wrestling with one right now, so wish me luck.