There is a machlokes Rashi and Tosafos whether people out of the Yerushalyim vicinity would keep two days of Rosh Hashana out of concern that Elul was a 30 day month. Rashi writes on 18a that they would only keep one day of Rosh Hashana. Rashi continues that the only reason that messengers went out for Tishrei is to prevent people from being concerned that they were keeping the wrong day for Y.K. and Succos, implying that they would not keep two days. Tosafos there and 18b disgrees and hold that they kept two days. However, the gemara on 21a makes it clear that they would typically not keep two days of Yom Kippur. When Levi visited he found that they were fasting on the wrong day, they weren't fasting both days. The gemara says that Rava would fast two day, and seems to be saying it as somewhat of a chiddush, implying that it wasn't actually necessary, just a personal chumrah. And Rav Nachman got upset when someone told him that the following day was Y.K. which would force him to fast a second day, putting his health at risk.
The gemara says that Levi pointed out that they were eating on what was established to be Y.K. in Eretz Yisroel, but could not formally testify to force them to fast that day because he didn't hear the beis din be mekadeish day 31. Rashi and Tosafos both explain that Levi was 100% confident that day 30 was not Rosh Chodesh and therefore it had to be on day 31, yet he refused to testify to force them to fast because of the technicality of not hearing the beis din formally declare day 31 as Rosh Chodesh (the difficulty with this is that the gemara 24a says we pasken like R. Elazar Bar Tzadok that on a me'ubar month, the beis din doesn't need to declare mekudash because it is automatic). Tosafos asks that since he knew that they were eating on Y.K. wasn't he obligated to stop them? Why is the technicality of not hearing the formal declaration of Beis Din sufficient to allow them to violate an issur kareis of Yom Kippur? Tosafos answers that since we say 25a that even if the Beis Din makes a mistake, it is binding, therefore it was considered Y.K. for the people of Bavel.
This Tosafos is very difficult to understand. The Turei Even already asks that the concept of אתם ואפילו מוטעין only applies to a circumstance where the Beis Din HaGadol in Yerushalayim messes up. The mistake of a local Beis Din couldn't possibly change Yom Kippur for their community? Furthermore, the gemara indicates that if Levi was able to testify because he heard the declaration of Beis Din, he would have been obligated to do so. Why would he have to testify for them, let them continue with their mistake since for them it would be binding as Yom Kippur? The Ritva explains that it was a believeability issue. There was a takana that one is not obligated to believe any testimony from someone who didn't hear the Beis Din say mekudash. Based on the Ritvah it makes sense that Levi didn't say anything because he would not have been believed on what he was saying since that was the takana. The approach of the Ritva would not apply the concept of אתם ואפילו מוטעין, so clearly Tosafos is not saying like the Ritvah.
The Chiddushei HaRan also assumes like Tosafos and explains that had Levi testified for them, they would be bound by the Beis Din HaGadol in Yerushalayim, since all batei dinim were kafuf to them. But without Levi's testimony they have independence and can also excercise the concept of אתם ואפילו מוטעין to make the prior day into Y.K. The Aruch LaNer also assumes that the local Beis Din must be able to also use this concept. Any place that could not possibly know better must have the right to be under the excuse of אתם ואפילו מוטעין, because otherwise when they find out that they ate on Yom Kippur they should need to bring a Korban. Furthremore, the Aruch LaNer explains that anyone who is able should be REQUIRED to fast 2 days out of safeik, which the gemara clearly assumes is not the case. Although the Beis Yosef cites the Yerushalmi that the heter for not fasting two days was a concern of sakana, the Aruch LaNer writes that one could not rely on this as a general rule, rather each person would need to figure out whether it would be dangerous for them to fast two days. Therefore he concludes that we must be able to apply the אפילו מוטעין even to a local Beis Din.
Although I think that the approach of the Aruch LaNer is the correct peshat in Tosafos, I think that the rejection of the Beis Yosef is incorrect. There are sometimes circumstances where if we would have people do something dangerous as a large group en masse it will statistically be a sakana for a percentage of the group. Even if by looking at individuals we would say that 95% of them would be able to fast without danger, but if the 95% would fast some would inevitably die, that would also be a sakana to permit eating on Yom Kippur. Aside from a concern that if we told the 95% to fast, there would inevitably be people from the 5% that would also fast and be a sakana, there would even be a sakana to the 95% because looking at it on a more global scale rather than as individuals, there is inevitable danger. A mashal to this is when a state hires 1000 workers to build a tunner or bridge. For each worker the chance of death may be small, but there is not question that someone will die in the process. In halacha, that may also qualify as sakana - v'tzarich iyun.