Flexible working is a blanket term for engagements which are different to the 9-5. In the professional sphere for mid to senior level hires, we've found flexibility occurs in three ways:
Part time: a defined flexible position - normally 2-4 days per week.
Full time with flex: 5 days per week with flexibility around start/end times and/or remote/office
Portfolio: 1-8 days per month - normally rotated alongside another portfolio position
This can be on an employed or self employed basis.
Normally this person is juggling family / personal responsibilities rather than another role.
It makes sense for remote working to feature here - at least 1 day.
Full time with flex
This too can be employed or self employed.
If it's the latter it makes sense this is for a defined period / project of say, 3-12 months to deliver a specific piece of work.
This is the expected working pattern for most UK professionals now and remote working when needed/wanted should be be the rule rather than an exception
This is best done on a self employed basis but it certainly can be long term as they'll be juggling another job and therefore within IR35 https://www.gov.uk/guidance/understanding-off-payroll-working-ir35
Ideally keep this office based (remote when needed) and work hard to get your processes set up here.
You can really hire top quality people in this role - truly self managing!