"Ode to Spot"
There is an unsupported remark in the Wikipedia article on Data indicating that visual effects artist Clay Dale had written "Ode to Spot".
However, according to this Q & A session with Brent Spiner, "Ode to Spot" was most likely written by Brannon Braga, a Star Trek producer and the writer of the episode "Schisms" in which it first appeared (and also the episode "A Fistful of Data", where the poem appeared again).
AUDIENCE MEMBER: Hi, Brent, do you still remember "Ode to Spot"?
SPINER: What, "Ode to Spot"? Um, Felus cattus is your taxonomic — wait — nomenclature, an endothermic quadruped, carniverous by nature. Go ahead (to the audience). You know what, I knew it by heart once. Um, it was a great poem actually. I think, er, I think Brannon Braga wrote that poem. Brannon's a wonderful writer really, and he's such a perverse human being and the stuff that he comes up with is always interesting. I really, really like working with Brannon. He wrote "Ode to Spot" which is, I think, Pulitzer Prize stuff. Yuh?
(Note: the heading of the web page for the Q & A session says "1977" when it should say "1997".)
"A Sunset Bloom"
According to the Memory Alpha article on anapestic tetrameter, the title of the poem beginning with
Then we sat on the sand for some time and observed...
is "A Sunset Bloom". (Anapestic tetrameter is the meter that Data used for the poem.)
Since this poem premiered in the same episode as "Ode to Spot", it would seem reasonable that Braga wrote this poem as well.