I think people would accept EVERY intrusion of privacy BUT only under the assumption that the user can choose who they're communicating with AND that the person they're communicating with can't peer into their life. The internet has made it clear that most personal boundaries exist out of fear. Fear that everyone will look at you differently if those secrets get out. People only let these secrets out when the "everyone" around them is a stranger. There are stronger boundaries such as those around traumatic experiences and criminal secrets though. Both of these are difficult to speak on even around complete strangers, but they're relatively rare. Only, AND, only if boundaries such as those two could be passed, would privacy become a major concern under the conditions that my first sentence set.
If SOO many people can have internet, phones, and credit cards in spite of the Snowden leaks(the fact that not only is EVERYTHING tracked, but that everything can also be pieced together back to you EVEN if your actions took place across dozens of different accounts, emails, addresses, or whatever, THEN further extrapolated on to give your schedule, who you're in personal contact with, and so on), then it's clear that the scariest aspect of intrusions of privacy is truly the backlash. Because, ultimately, who cares? How many people do you know that were jailed due to all this data the government's been harvesting? Odds are zero.
For your scenario, so long as the chips aren't used to aid in the arrest of low-profile citizens, I doubt anyone would care much. To take it further, I doubt any government would just say "these chips are mandatory." They'd probably avoid most of the backlash by getting middle-men to say "these chips are mandatory." For example, they'd incentivize businesses, sites, and internet services to have it as a requirement. Whether that incentive is something positive like putting/leaving money in their pockets, or negative such as taxing them, it doesn't matter. How many people would have these chips if they were practically mandatory to access half of the country's jobs or the next big social media platform? This is part of why phones blew up to extent they have. People that don't have phones are the minority, and people that don't have chips would be one too. Most people would passively accept.
