On the other hand, having an sign-in-less "Explore" type experience may actually hurt your signup conversions. Users get a feel for the app, decide that they'll sign up later, close the app, and never come again.

I might be hallucinating but I think Instagram used to have an "Explore" screen with interesting pics pre-signup, and in the latest versions they got rid of that.

You may certainly be correct, but part of your viral coefficient is likely driven by continued usage of your application. As such, I guess there are two ways you could go about it:

1. Spam user's social network on sign up; 2. Send out periodic blasts about progress/offers as the user continues using the app.

The first method will likely net you terrible reviews and negative sentiment, while for the second, an exploratory mode wouldn't really hurt you. The key would be designing exactly when and how the sign up should be. For example, maybe after 5 days or 5 app opens (whichever is more), the app loses functionality until you connect. It'd be different for every app. For games (or gamified aspects), you could just incentivize sharing.

On the iOS platform, you want to be very careful about timing when you ask for the push notification permission. which you'll need it for scheme 2.

One newbie mistake on iOS is to ask for notification permissions right after install. This is very similar to your point about signing up: You have to build some trust with the user first, then ask to notify them.

I meant socially, not for push notifications. Since, after all , the goal is to engage more users. You don't need push notifications here at all.