Short of following everyone on the other instance, the "local" view on each likely has people that you aren't following.
You can't get the "what's the buzz on the other one?" from viewing the home, local, or federated views.
Additionally, the discoverability of your content on one server depends on it getting followed and people watching the federated feed rather than local or home feeds (and when people on the site are following other sites, your content is difficult to be discovered).
The federated feed is a firehose that goes too fast for discovery.
The local feed is domain/subject/theme appropriate and where most people likely browse content.
Your content would only show up in the local feed of the other if someone on that other one boosted it which implies that they discovered it on the other instance and followed your account there.
If you post something math related on the programming instance, the only way for it to show up in the local feed of the math instance is for someone to boost it there. Short of having someone boosting everything that you want to say on the other topic on the other instance, it might be less friction to maintain multiple accounts.
That's not the way that it "should" work, but that's the way that people are using it. The "town square" of the local feed is where interesting things happen.
Eh, that comes down to how you choose to use the platform.
Speaking for myself, early on I used the local and federated feeds to find interesting people to follow, but once my followed list was built up, I found I rarely spent time in those feeds.
Personally, I'd say if folks are using Mastodon in the way you're describing, they'd be better off using Lemmy or kbin, which are centered around communities containing topics, as then you can just follow those communities rather than following individual people.
When groups that formed around hashtags on twitter moved to mastodon, they did so as a group. What was #footwitter is now local on foostodon.social. For example "energy twitter" (where people discuss power grids) https://twitter.com/hashtag/energytwitter has had a partial migration to https://mastodon.energy/public/local
When people migrated off of Twitter, and mastodon.social wasn't accepting new registrations, and "taking over" a random one was seen as impolite, someone stood up an instance and said "hey, we're over here" and local will be encouraged to have mostly energy related information.
If you were following two different hashtags on twitter, and they're now two different mastodon instances, to have a similar experience you need to be able to follow local on each and push content to either such that it shows up in local on the proper one.
With the way that mastodon is organized, that's not an option and the way to get to that similar experience is to have an account in each.
Would Lemmy or kbin have been a better choice for such? Maybe. However, for a twitter like experience, they aren't a proper substitute.
You can't get the "what's the buzz on the other one?" from viewing the home, local, or federated views.
Additionally, the discoverability of your content on one server depends on it getting followed and people watching the federated feed rather than local or home feeds (and when people on the site are following other sites, your content is difficult to be discovered).
The federated feed is a firehose that goes too fast for discovery.
The local feed is domain/subject/theme appropriate and where most people likely browse content.
Your content would only show up in the local feed of the other if someone on that other one boosted it which implies that they discovered it on the other instance and followed your account there.
If you post something math related on the programming instance, the only way for it to show up in the local feed of the math instance is for someone to boost it there. Short of having someone boosting everything that you want to say on the other topic on the other instance, it might be less friction to maintain multiple accounts.
That's not the way that it "should" work, but that's the way that people are using it. The "town square" of the local feed is where interesting things happen.