This has been discussed before, but yes, if you are working remotely, you have some options, and can legally "work", even in Europe, depending on your companies policies.
The key points are: (US for this example)
1. you are a US Citizen and resident of the US.
2. Employed by a US Company, doing work for your US location, and paid in US funds at a US Bank.
3. Are not seeking residency or a long stay visa for the country you will be in.
Contrary to public belief, the Schengen 90/180 day Visa Waiver and the UK "Up to 6 Month" entry allowance allows for business (work) activity to be done. This includes business meetings, seminars, audits, working with subsidiaries, customers, and suppliers; giving lectures and other educational activities, basically a whole host of activity that is "work" but short in term on a visiting basis and as an extension of your job in the US.
Same with remote work, you are simply doing your normal job, meeting the conditions above, and in the eyes of the government, doing no harm. Like living in another State, or spending a few months in another country, you are not resident, so you do not have tax liability or rights as a residence.
You should be able to find this information for about any country, but as an example, here is a link to the UK Government site that helps you determine the need for a Visa and allowable activities: https://www.gov.uk/check-uk-visa
By inputting pertinent information, you can get to an appendix that discusses allowable business related activities: