Depends on the server, and the channel on the server. Depending on the channel, you might get a handful of posts a week while the #general channel might be a non-stop firehose (at least on the more social Discords).
As to how it’s the closest alternative, mainly in that the atomic unit around which Discord is organized like Reddit is the community. On Reddit it’s subreddits and on Discord it is servers and it has gotten complex enough that for server owners Discord is slowly becoming a build your own social network toolbox.
Instead of the sidebar, each channel has pins. You can have an arbitrary number of channels for different topics, and you can even have forum channels where at the root level of the channel are forum topics. Once you’re in a server, searching it is trivial to see what people have said in Subject X in community Y, although this isn’t as good as say, searching all of Reddit is and you have to find a decent community to be able to search it.
Also a lot of subreddits are already running a Discord server on the side with the same or a similar mod team and have been for years. Doesn’t quite resolve Discord’s server discoverability issue, but Reddit doesn’t make the worst springboard for figuring out Discord either. If you support any Patreons or have similar memberships elsewhere, there’s a decent shot you already have access to a number of private members-only Discords which has kind of been my springboard into actually using my Discord account this past year.
But you know, it is very different. I don’t think it’s nearly as perfect an analogue to Reddit as Reddit was to Digg for Digg users, but it’s probably the closest to what Reddit already is.