Is it permissible to say, "So-and-so narrated to us," when hearing Hadiths from Shaykhs via screen or recordings? And is it permissible to cite the isnad (chain of narration) in this case? And is it permissible to say that I am a student of so-and-so if the learning was through his tapes and books?

If someone hears a hadith (definition) or a discourse from a shaykh via a screen or a recording, it is more appropriate for them to say: "I heard so-and-so shaykh on such-and-such tape/channel saying such-and-such." This is more precise and honest than saying: "So-and-so shaykh narrated to us," because the latter is a technical term in hadith terminology indicating direct audition in his assembly. Similarly, the claim of "discipleship" to a shaykh is usually applied to a direct student who accompanied him closely.

A student should not present an isnad (definition) (chain of narration) for a hadith they heard from a shaykh through modern media, because they are not a muhaddith (hadith scholar) narrating hadiths with their isnad, and the hadith is usually available with its isnad in books.

Finally, listening to recorded lectures is beneficial for acquiring knowledge, but it does not substitute for sitting with shaykhs, traveling to them to attend their lessons, and learning from their manners and conduct. This is what is called "becoming their disciple."