"Is there any evidence that tweets have meaning? I could easily imagine a world where posters fool themselves into thinking the words they wrote contained their intentions, while readers fooled themselves into thinking they had uncovered those intentions."
As with all natural language, production and interpretation of poems is subjective. Poetry is just an attempt to use natural language without the constraints of prose, in order to accomplish things that are not possible with prose.
Some poets may have a goal to convey a particular idea or emotion. Others may just want to create something moves the reader. Still others may not care about the reader at all. All of these things are okay.
Actually, that speculation about tweets sounds pretty realistic to me...
But seriously, I think we can all agree that among the literary arts, poetry is the most likely to be accidentally uninterpretable. Due to its cultural context it is also the area where uninterpretability is the most likely to be accepted. This creates an environment where you run a serious risk of developing a culture of meaninglessness.
Some poetry serves the "higher" meaning of discussing political ideas like Claudia Rankine's Citizen: An American Lyric.
Some poetry is made to make you laugh like Billy Collins' Another Reason Why I Don't Keep A Gun in the House.
Some poetry is purposefully inscrutable and difficult because the author wants you to work to understand them. A good example of this might be r-p-o-p-h-e-s-s-a-g-r by E.E. Cummings.
Each of these examples is meaningful in its own different way. I think trying to decide what has meaning is hard because you might automatically discard a work of art that is "just for fun". Isn't play meaningful?
as everybody else in Italy, I had to study the Divine Comedy (which is an incredible work, and I'm glad I had to), and part of doing that is learning multiple interpretations of a single line, variations of people who have read and re-read the work over centuries. It really easily convinces you that, if the art itself can have a given meaning, still many interpretations are bonkers and there might be no hidden meaning at all.
I am sure the same applies, to e.g. William Blake.
As with all natural language, production and interpretation of poems is subjective. Poetry is just an attempt to use natural language without the constraints of prose, in order to accomplish things that are not possible with prose.
Some poets may have a goal to convey a particular idea or emotion. Others may just want to create something moves the reader. Still others may not care about the reader at all. All of these things are okay.