You don’t need to purchase the ARD product here, as the screen sharing client that’s integrated (free) with macOS will be just as much a hassle as the paid ARD client app; as will be remote-network access using ARD.
This requirement you’re being a problem involving networking, firewall pass-through or firewall VPN servers, and of VPNs, or around the related decisions of exposing your target network to the riffraff; of whether to use a VPN to connect into the target network either via a VPN server in the firewall, or (less desirably) to open up port forwarding on your firewall for the VPN or (even less desirably) for screen sharing itself.
Screen sharing client and the ARD client and the integrated screen sharing server in macOS do not provide this pass-through. They’re intended for local-network-only, or with hosts with public static IP addresses.
To work here, the Apple screen sharing server—this screen sharing server is used for both the free screen sharing client, and the ARD client, and is integrated into macOS—would need to expose the target system for remote access beyond the local network. The server doesn’t drill through and doesn’t register your server—the Mac system you’re trying to access, here—for remote access.
Or somewhat differently phrased, neither screen sharing nor ARD will resolve remote private hosts on firewall protected private networks using private addressing.
To expose that access, you’ll need to make some configuration changes, and potentially upgrade your firewall, and you’ll likely also need to configure dynamic DNS for a residential IP address or configure static DNS if you upgrade to a static network connection; to an ISP service tier with a fixed IP address for your firewall.
How much do you want to learn about IP networking, firewalls, VPNs, and server security? I and Antonio Rocco and others here can certainly help with this, but we’ll need more info on your network and your budget and your security requirements. And you’ll be learning more about IP networking, too. (And there can be added wrinkles, as some ISPs can block some in-bound network traffic.)
If you don’t really want to learn about that and about IP networking... There are vendors which have clients that can drill through into remote networks, though most are commercial offerings. Examples include LogMeIn and alternatives.
Or you migrate and host the data of interest outside your firewall. Out to iCloud, for instance.
You can log some feedback with the folks at Apple, asking that the integrated screen sharing server be updated to allow remote access. That’d be useful for this case, and various others. Right now, that remote access requires somebody on the target network set this up either in the firewall or using an add-on remote-access app such as that of LogMeIn or such, or required somebody on the target Mac initiate screen sharing via Messages. Product Feedback - Apple