Japanese shake hands..

Japanese shake hands rarely with each other. When they greet or say goodbye to other they might bow at the same time, depending on the circumstances. But they wouldn't ordinarily shake hands. Of course they know that westerners shake hands. And many of them expect Japanese to. In that case it's quite all right. It's just that in general Japanese don't touch each other much in public.

Are western-style greetings offensive?
Not really. Japanese know that they've got to adapt somewhat to different customs. It's sometimes a bit awkward for them at first. Japanese are not used to the bear hugs and kisses with which westerners often greet each other. But they manage.

Many Japanese who are around westerners a lot don't bow to them at all. Others who are not used to foreigners may bow because it's automatic. But Japanese don't usually expect foreigners to bow in return.