How do you job search while employed without it getting out?
By controlling the three channels through which it leaks — your employer's systems, your professional network, and recruiters — and by recognising that the risk is real but manageable, since most people who are found out are found out through avoidable mistakes.
The specific leaks and how to close them:
Company devices and accounts. Never job search on work equipment, work wifi or a work email address. Employers can lawfully monitor work systems, and browsing history, email and file access are visible. Use a personal device on a personal network.
Professional network profiles. Suddenly updating a dormant profile, adding a photograph and gaining connections is a recognisable pattern. Turn off activity broadcasts before editing, make changes gradually, and consider the "open to work" setting — the recruiter-only version is not visible to colleagues, while the public badge is.
References. Never list a current manager without permission. Use former managers and colleagues, and tell prospective employers that your current employer may be approached only after an offer — which is normal and accepted.
Recruiters. A recruiter submitting your details speculatively to your own employer is a genuine risk. Tell them explicitly which companies they may not approach, and ask before each submission.
Interview logistics. Frequent unexplained absences, dressing differently, and long lunches are noticed. Use annual leave, schedule early or late, and take video interviews from home.
Discretion with colleagues. The most common leak is telling one trusted person.
Social activity, including likes and follows of competitor accounts.
What you are actually entitled to:
Looking for another job is not misconduct. You may be contractually required not to compete or solicit while employed, and searching is not a breach.
Time off for interviews is not a statutory right in most cases — except where you have been given redundancy notice, when reasonable paid time off to look for work is a legal entitlement.
What to do if it does come out: be straightforward rather than defensive, and expect that in some organisations it changes how you are treated regardless.
General information, not legal advice.