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I know a little bit about Judaism and a fair amount about hearing aids. I have been asked about the rules regarding the usage of hearing aids on the sabbath and I have no idea. In particular, the person wants to know if an analog aid with the controls built into the device is different from a digital aid with a remote control for controlling the settings.

Tzomet is an Israeli institute for Science and Halacha (Jewish law). They research modern devices and see how they can be used/tuned to be used with observant Jews. They have some relevant thoughts on hearing aids (here)

In summary, the main halachic requirements for using a hearing aid are
to have appropriate long-lasting batteries and to turn it on before
the beginning of Shabbat.

The rabbis of recent times have decided that if the hearing aid is
turned on before Shabbat (that is, the main switch is ON) it can be
used on Shabbat. In addition, one is even permitted to change the
volume of the device using a tiny wheel or a straight-line control.
(However, the volume may not be adjusted on Shabbat if it is
controlled by pressing a button, similar to a television remote
control.) Obviously, it is necessary to have batteries that will last
for the entire Shabbat, and this is possible with the very low power
consumption of modern hearing aids and the newest highly efficient
batteries.

The "setting" of the device should not be changed manually on Shabbat
(if this is possible with this type of hearing aid). However, the
latest most sophisticated "digital" devices have internal computer
controls which automatically change the parameters of the hearing aid
depending on the environment (such as sound level). We allow such
hearing aids to be used, and we do not see any halachic problem even
when the user moves from one place to another, causing the internal
parameters to change automatically.

It is preferable to use hearing aids with a condenser type microphone
(in which case there is a weak current on all the time) and not a
dynamic microphone (where a new current is created when there is a
sound, similar to a bicycle dynamo).

In addition, the Shemirat Shabbat Kehilchata (a standard text of Shabbat laws, 34:30) writes that hearing aids may be used if turned on before Shabbat, that volume can be adjusted if the current doesn't make a wire glow red or stop glowing. He cautions of not carrying any hearing aid part in one's pocket outside of areas with an eruv.

Of course, consult a rabbi
before implementing anything you learn here.

Welcome to Mi Yodeya.

As my son wears cochlear implants, I have been reading quite a bit on the usage of hearing aids and implants on Shabbat. There is quite a diversity of opinions in this area. However, Rabbi Elisha Sandler's article is about the most comprehensive that I can find on the web. I know Rabbi Sandler, so if you have any specifics that you need to inquire, ping me, and I can see if I can contact him. (I don't recall if he included contact info in the article.) I'm excerpting the conclusion / summary. You should read the entire article to understand the issues, and make your own educated decision. I would also suggest showing this to your local rabbi, so that he can gain some perspective and offer advice.

He doesn't seem to differentiate between analog and digital aids regarding any of the issues.

There are differences of opinion regarding the initial activation of a
hearing aid and the replacement of its batteries. Both should be
avoided. If this is not possible, there are significant authorities
who permit these actions (preferably by a non-Jew or in a backhanded
manner). One may speak directly to people, especially children,
wearing hearing aids and certainly to an audience that is only
partially composed of such individuals. The handling of a functioning
hearing aid is unanimously permitted if this is done to use the
hearing aid itself or the place upon which it was placed. Handling a
hearing aid for any other purpose will be subject to how its muktzah
status is categorized. Hearing aids may be worn in a public domain
according to many authorities since they are not regarded as a load
and will not come to be carried.

There is also a pamphlet called Torat Hacheresh by Rav Mordechai Shuchatowitz who is connected with Our Way, the deaf division of the OU. It's an excellent pamphlet explain all sorts of halachot for the hearing-impaired. If interested, contact me via chat.