With the new website, I've never received any kind of notification for game updates, forum replies, or messages. Has anyone else noticed this issue? I'm using the latest version of Firefox.

Yeah, there are no notifications in the new website, for now. Gog are working on it, but we have no idea when we'll get them back.

In the mean time Barefoot Essentials replaces this functionality.

haha. Hope springs eternal.

OP, besides Barefoot Essentials, you can also use Galaxy if you want. At least that's what I heard.

Soon™

GOG promised they were working on fixing it.

Here's the Link to it's Forum
<span class="bold">Barefoot Essentials</span>

Somebody not to far away, also made this handy companion for it.
<span class="bold">Adalia Fundamentals</span>

Which also adds back some lost features, and for me the essential Manual sorting, GOG removed when Galaxy appeared.
It was manual sortings removal, that led to me discovering these must have user scripts.

Both are essesntial User Script additions to my GOG experience, with many removed features, and some new ones, you don't need until they are available, and then you can't live without them.

They require a user script extension, as the posts linked explain, I use Greasemonkey in firefox, and they work great with it.

We're past soon™, it's eventually™ now.

The website is discontinued and in the process of archiving. Please use GALAXY, our optional client, to get the most convenient GOG experience.

I've just installed Barefoot Essentials, but how can I tell if Notifications are enabled? Will it show an icon on the navbar?

It should give you a red box with a number in it in the navbar.

Moving quickly to Possibly Before The Heat Death of the Universe™?

I'm still not seeing that on the navbar. Is there a way to enable it?

I think it's enabled by default, actually.

Even though I'm getting replies on the forum, I'm still not seeing a red indicator. Do I need a certain version of Barefoot Essentials?

Yes, update BE to the latest one.