Do you need a certification?
There's no single global answer, and the differences between countries are bigger than most people expect. Check the rules where you live before you accept money from anyone — this is the one step in this guide you shouldn't skip.
At one end, coaching adult amateurs privately isn't a regulated activity at all: no licence exists and none is required. At the other end sit countries where coaching for payment is tightly regulated. France is the clearest example — under its Code du sport, only holders of a recognised, nationally registered qualification may teach or coach a sport for remuneration, even occasionally or as a side activity, and coaches must additionally register with the authorities to obtain a professional card. Coaching for pay without meeting those requirements is a criminal offence there, not an administrative oversight. Several other countries have their own versions of this.
Between those extremes, most places regulate specific contexts rather than the activity as a whole: clubs, federations, official competition, and work with minors typically require formal qualifications even where private coaching of adults doesn't. Insurance providers may ask for credentials too.
Two further things worth checking locally: whether remote or online coaching is treated the same as in-person coaching — in some jurisdictions it clearly is, in others it's genuinely unsettled — and whether you need to register as self-employed or a business before invoicing. The answers vary enough that a few minutes on your national sports ministry or federation website is time well spent.
What's universal is that a certification does two useful things regardless of whether it's legally required: it fills gaps in your knowledge, and it gives athletes a reason to trust you before they've seen results.
Broadly, there are two routes. National federation pathways — the athletics or triathlon governing body in your country — tend to carry the most weight in club settings and competitive sport, and are usually the route that matters if you want to work inside an official structure. Independent certifications such as those offered by UESCA or RRCA are typically available online, are specific to running rather than to athletics as a whole, and suit coaches working remotely with amateur athletes.
The choice mostly comes down to one question: do you want to work inside a club or federation structure, or independently with recreational runners? The first points toward your national body, the second toward a specialist certification.
Whichever route you take, look into liability insurance for coaches. It's usually inexpensive, and you're working with people's health.