I'm not going to comment on your specific situation, but I will make a few observations:
* If you maintain an Upwork account, there is no way to GUARANTEE that your employer won't find out about it. Keep in mind that this is on the Internet. It is worldwide. Upwork's purpose is to maximize the ability of clients to find freelancers from all over the world. So the purpose of this site is essentially the opposite of privacy.
* There ARE privacy settings. These limit the extent to which non-Upwork users can find your Upwork profile. And they limit the extent to which common search tools such as Google will be provided with your freelancer profile. But that doesn't mean that your profile will be truly "private." It IS still on the Internet. Nothing on the Internet is 100% private. What if your employer goes looking for a freelancer on Upwork, posts a job posting, and you apply to it? If that happens, then nothing about your Upwork profile page will be hidden from your employer. Well, couldn't you just not apply to your employer's job posting? What if your employer is using Upwork while sitting next to his wife, who is using HER client profile? Or his cousin? Whose name you don't know? Or what if somebody clicks on to you profile land then takes a screenshot, and sends it to his email list? Or posts it on his Facebook account? And your employer sees it there? I could list endless scenarios.
* Your employer is almost certainly NOT going to see your Upwork profile. Even if you use 100% public settings. Your employer doesn't spend time googling you. Your employer probably isn't using Upwork at all.
* Many companies have clear policies that allow their employees to freelancer, or do other work on the side. Some companies have policies different from that. If you think your employer wouldn't approve, one approach you could take is to not use Upwork.
As Preston said, nothing online is guaranteed to stay private. That said, we can exercise some control. I don't have an employer to worry about but I keep my profile set on Upwork Users Only. UW clients can find me but not just any old body out on the Internet.
BTW, remember this forum is public.
The invisible and online status is not related to your profile visibility. If you do not want to be seen by your client as active while logged in to your Upwork account, then you can set your status to invisible. If you like to show as online, then you set it to online.
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