My $0.02.
I am a current student from the July 2020 batch in the Singapore campus. Have attended quite a few recruitment presentations (targeted to the 2019 December class, but we are also allowed to attend). I have also worked in Malaysia for te 2.5+ years before joining INSEAD with zero local language knowledge. Here's a summary of what I have gathered based on my experience in the region and interactions with the recruitment teams so far, especially for SEA countries:
Consulting: MBB doesn't hire foreigners in Singapore (maybe a couple here and there) due to government restrictions. However, they hire foreigners in Malaysia, Indonesia, and the Philippines. It is an advantage if you have local language skills, but it isn't a necessity. Other consulting firms do hire foreigners in Singapore. Numbers are on the lower side, but its better than MBB.
IB/PE/VC: Foreigners are hired in Singapore, but numbers are a bit limited.
Tech: Big tech as well as startups hire foreigners in Singapore. They aren't much affected by government regulations as they maintain an acceptable ratio of foreigners vs. locals. Indonesia is another hub for tech startups, and foreigners are hired (local language skills is an advantage but not necessary). Typically, roles based out of Thailand or Vietnam require local language skills.
Other Corporate roles: Some of the big players in the healthcare industry do not hire foreigners in any location. So you can only apply for a location where you have work authorization. Most other roles are open for foreigners and do not require local language skills for Singapore, Malaysia, Indonesia, Philippines, or in some cases even for Thailand.
All roles in China require fluency in Mandarin. Haven't been to any event where a role was specifically offered in Hong Kong. So can't comment on the requirements there. But based on experience I'd say for finance roles just English would suffice (have a few friends working there with no Mandarin skills).
Some global conglomerates from Japan and Korea hire foreigners and do not require local language skills. However, the location for these roles may not necessarily be in Asia. Some of these are rotational programs where you change your city every one year or so.
Hope this provides some clarity!