It's largely trial and error unless you find existing data points. Sometimes you only need to use the specific website for a different country as noted above; other times VPN is needed or helpful. I saved a decent chunk on SCL-IPC-SCL tickets using the LAN (at the time) Chilean website. But often I've struck out, too.
Also the embedded Google Translate feature in Chrome can be a big help when browsing other-country sites. The basic search fields are pretty much self-explanatory in any language, the translation can help with reading T&Cs, etc. But the larger foreign airlines may even have an English language option on their native country site as in the above poster's example.
A minor side note: Although not terribly common, you do have to watch for fares that are restricted to residents of a given country. LATAM has Ecuador resident-specific fares, for example (used to have the same for Peru but I understand those are no longer offered). Generally you will see a warning on the website or a note in the fare conditions to that effect. Again, it's not widespread so don't be overly worried but does exist in some places. If in doubt contact the airline. (Though usually you can find discussion of these instances in the relevant airline forum here, or on TA or similar forums, too.)