The paper begins at your old school
Your diploma was made by the school that sat you the final exam, and signed by the head and the board chair. That is why how to apostille a Diploma starts in the school office, not at a state counter. Ask them for a clean copy or a short letter of award, both of which carry a live signature.
Who adds the stamp
To apostille a Diploma in Italy you deal with the Prefettura, the branch of the state that sits in each provincial capital. A school is a public body, so its paper follows that route. The Procura della Repubblica, based at the court, only takes acts made before a judge or a notary. So a notarised copy of your diploma would go there instead.
Fees and timing
The apostille cost for a Diploma in Trieste is a €16 tax stamp for each paper, and some papers are let off it. The school may ask for its own charge to print a copy. Nothing here sets a waiting time, so allow for the school and the counter in turn. A lost diploma adds a search on top.
Why a college abroad wants it
School leavers apostille a Diploma in Trieste to enrol at a college abroad, to join a trade scheme, to sign up with an army or police force, or to back a work visa. Italy came into the Hague Convention in 1978, so a single stamp serves member states. The stamp proves the signature, and nothing about your marks.
Language notes
The legalisation of a Diploma in Trieste is about the seal, so the words stay Italian. A college abroad that reads no Italian will ask for a sworn translation of the award and the subject list. A Tribunale stamps that translation after the translator swears to it. Old handwritten diplomas are hard to read, so give the translator a sharp scan.