Ksemendra's text the Kavi-kanThA'bharaNa is an important manual of kavi-zikSA. He lays out rules about how the aspiring poet should read the compositions of past masters, cultivate an ear and a palate for good poetry - both in form and content; and prevent the malodour of sanctimony [and VedA'bhyAsa] from wilting the sprout that may have started sprouting in his heart. I am unable to recollect the verse.
It is interesting how he recommends not just high Sanskrit poetry, but also poetry in Prakrt and the vernaculars [please note his use of the term deza-bhASA to refer to the latter] -
Rajashekhara too has a chapter on training in poetry. There is 13th century manual of poetry, Kavya-kalpa-latA by Arisimha, completed by Amaracandra yati.
John Cortin his book 'Jain Communities in Indian History' says something to the effect that jaina writers dominate the scene in writing (systematic) texts on poetics and rhetoric, especially those that emphasise kavi-zikSA, or training in the art of poetry.
In the modern context, as Prof Deshpande pointed out, students are encouraged to try their hand at versifying - not just to get a grip over metric templates but to develop a sense for idiomatic language, and to get a nuts-and-bolts picture of the various rhetoric and poetic elements of language.
Colleges and schools conduct competitions for the composition of such poetry. A session of impromptu poetry recitation is often part of the vidvat-sadas - symposia hosted by Religious Mutts.
The central Sahitya Academy has restarted the journal SamskRta Pratibha, which had a long run in the hands of Prof V.Raghavan. It is a bit disheartening to see some of the poetry that is featured in it now - the verses are blank both in metric structure and poetic content.
Radio and National TV [Doordarshan] sometimes host gathering of poets, where they are encouraged to compose and discuss verses on particular topics. I too have had occasion to participate in such events, and share my feeble attempts at Sanskrit poetry.