How does that work if a pool sends out regular newsletters and writes "discuss any health issues with your swimming teacher before class"
Such use would not be regarded as "use as a trade mark" under s120 of the TM Act that deals with infringement. You can use TM's in many different ways without infringing them.
Descriptive or comparative use is but one example. i.e. "Headlight is compatible with Ford Falcon model years ...". The TM's FORD and FALCON are being used, but not as a badge of origin. You are not saying, by referencing these words that you are the source of FORD or FALCON products.
In the present case - its slightly different. A reference to "your swimming teacher" might literally be a reference to the person who offers swimming lessons. So that even if you did get a TM registration for YOUR SWIMMING TEACHER, mere use of the phrase will not infringe it.
As Rob mentioned above, it is unlikely that you would get a registration for SWIMMING TEACHER for swimming teaching services as the words are totally descriptive. Any swimming teacher should be entitled to use the phrase "SWIMMING TEACHER" in their business name.
Descriptive TM's are terrible in any case from an IP perspective as you can use them "descriptively" without getting stopped by the TM owner. That means that your competitors can get a LOT closer to your business name without infringing.
The best TM's to register have NO CONNECTION to the services that are provided under them as any person trying to copy the name will be clearly doing so to ride off your reputation. They will not have the descriptive use excuse to fall back on.
Hope that helps!