When I try to set a skin with a resolution of 1024x512, and 606 KB weight (20 times the maximum limit for local export, with a maximum of 30 KB), then on the second client (connection via "Open to LAN") instead of the skin, above player displayed:

Model failed to load: javax.net.ssl.SSLHandshakeException: sun.security.validator.ValidatorException: PKIX path building failed: sun.security.provider.certpath.SunCertPathBuilderException: unable to find valid certification path to requested target

GitHub Gist was used for export, on 1 client side (who uploaded) the skin is displayed normally, Gist is available, the connection is stable. As for the error, I only found on the Internet that this is due to “The Java application cannot create a trusted TLS / SSL connection. Perhaps the application is trying to connect to the server via HTTPS and the server is using a self-signed certificate or a certificate that is not accepted Java node."

I am more than sure that I am doing something wrong, but so far I have not found a solution for myself. In addition, this problem exists for me on both 1.7.10 and 1.12.2, so again, most likely it’s me doing something wrong, but neither in “Exporting” nor in any other opened issue I didn't find anything

You have to install an up-to date Java 8 version and use it to play the game.
The default Java version that is shipped with the Minecraft launcher is very old and is missing the certificates to connect to most modern websites. (Mojang ships a Win 7 java build due to broken intel drivers on older PCs.)

I'm using prism launcher with a custom java version - jdk1.8.0_351. Is this still a problem? If so, which version should I use, 8u391?

Quite strange, but this is true. Upgrading from jdk1.8.0_351 to jdk1.8.0_391 solved the problem, thanks!