I'm not a hadith scholar and I am new to this field. This is what I've read so far.
There are a number of hadith categories in Twelver Shia hadith science with respect to the reliability of the narrators:
Sahih: a hadith with everyone in its chain of narrators reliable and Imami. There are also subcategories to this that I refuse to name.
Muwasaq: a hadith with everyone in its chain reliable even though NOT Imami
Hasan: a hadith with connected chain of narrators with everyone in its chain "praised" and Imami.
Qawi: a hadith with chain of narrators with everyone in its chain Imami but without any report praising or endorsing the narrator.
Dhaif: a hadith with individuals in its chain of narration failing all of the above criteria.
Based on what I've read, while there's a minority who believe that all narrators in a hadith must be Imami, the general consensus is that the ultimate criterion is whether the narrator has the general requirements of honesty, integrity and maybe competence so that you can make sure about the authenticity of the report.
There are actually Shia hadiths that support trusting a verifiable report regardless of the narrator. For example, from Imam Jafar us-Sadiq it is quoted: "Take from them (Sunnis) whatever they narrate, and leave aside their opinions". Here "to narrate" obviously means facts that they faithfully report not allegations or claims or interpretations.
And there are a minority of scholars who argue that even if there's hadith/narration with a Sunni narrator nothing about whose reliability has reached us, the hadith can be accepted by ijtihad of ulama after examining the content or the circumstances of the narration.
However, any and all hadiths must not be contrary to the verified hadiths that have been related by Imams of Ahl al-Bayt. This rule presumably can override even reliable narrators for no narrator other than Imams of Ahl al-Bayt are considered infallible and totally immune to error. Therefore, the reports of even many Sahaba are questioned due to their conflict with the reports by Shia Imams or Imami companions and narrators. After all, "Justice of Sahaba" is also not a given in Shia belief.
And then there are some complications resulting from theological positions. One standing issue in Shia Islam has been accusations of "ghuluw" (committing exaggeration about Shia Imams) against certain certain hadiths or Imami narrators. Now here depending on what a scholar considers "ghuluw" a hadith or a narrator may be accepted or rejected.
In my online search I couldn't find a notable non-Imami hadith narrator who is considered reliable by Shia scholars. But there are certainly such individuals. Among famous narrators/scholars, I have the Shafi'i Mu'tazili Ibn Abi al-Hadid in mind, the author of Sharh Nahj il-Balagha, whose commentary is often cited in support of Imami beliefs.