A Judicial Assistant, under general supervision, provides clerical and technical work through the acceptance, examination, and processing of legal documents; prepares and distributes court calendars; files and retrieves documents and case files; and performs related duties as required.

специалист судьи: Подготовка судебных дел к процессу‚ назначение дел к производству‚ ведение корреспонденции‚ участие в судебных процессах‚ проработка решений и других судебных актов‚ выдача исполнительных листов.

это не помощник судьи в строгом смысле этого термина.
если помощник судьи - deputy jugde, с соответствующими функциями, то этот термин использовать нельзя. В процессуальных кодексах "специалист судьи" не встречается, судя по всему - это какая-то новелла, связанная, скорее с штатным расписанием суда как организации, а не с выполняемыми функциями. Думаю, можно передать смысл фразой aide to the Judge или же вообще термином clerk.

Обычно они выдают (или рассылают) разные копии и др. документы.

law students who have taken their degree and want to become judges, have to fulfill several duties at court over a period, so to speak, of apprenticeship the duration of which varies, but mostly it´s 3 years or so. Being well conversant with law, they are experts but no judges yet. They alleviate the burden of the latter, thus are more or less their aides or assistants.

(Job description excerpt)
Position Title: Court Specialist II
Grade: 120
Status: Regular Full-Time / Nonexempt
Reports to: Supervisor and / or Manager
Nature of the Work:
This is intermediate level, office and clerical work, which follows Florida statute, rules of
procedure, and well-established office practices. These tasks are specifically designed to support
the judiciary, members of the bar, federal, state, and county agencies, the general public and
interacting divisions. Employees in this class must be able to communicate effectively with
members of the legal community, judiciary, general public, and other work related agencies.
Work is subject to close supervision and reviews by a superior for productivity, accuracy, and
completion of assignments will be conducted periodically. Detailed instructions are received for
new assignments, with regular routine assignments performed more independently as experience
is gained.
Illustrative Tasks:
May be responsible for the timely and efficient gathering, reviewing, sorting, validating,
certifying, completing and distributing of various reports, legal documents received for filing,
scheduling hearings, recording, collection of fees, auditing, and/or further processing which may
include forwarding documents to interacting departments or agencies utilizing automated and if
applicable manual procedures..........