One method would be to measure various air quality parameters before starting to use the plant remediation unit and then successively over the course of weeks or months during its use. The key points here would be that you would need (1) to know what the most appropriate air quality parameter or indicator would be for your situation (e.g. formaldehyde, carbon monoxide, particulate matter, benzene, etc) and (2) have a reliable instrument and method for measuring that air quality indicator. If you're just interested in getting a sense of whether or not it's working at all, a qualitative or somewhat imprecise or inaccurate method may be perfectly fine.
Another potential method would be to keep a log of health symptoms starting before use of the plant remediation unit and continuing throughout its use. The primary benefit of this method is that it focuses on your health, which is likely the underlying purpose of using a remediation unit. The primary drawback of this method is that it can produce the "placebo effect" where you may trick yourself into thinking it is working when it's actually having no effect, or you may trick yourself into continuing to feel sick or fatalistic, but it's actually working quite well. You could perhaps invite a friend who doesn't know about your plant remediation unit plans to visit for prolonged periods of time and ask them about how they feel, but again, that is highly subjective.
Any other ideas? Has anyone heard of visible changes in the plant roots? There are some plants that could be planted in an indoor garden that may provide insight due to their sensitivity to airborne contaminants. Perhaps planting some of these at various points in time, and monitoring their health and vitality (or lack thereof) could be indicative of the air quality? Here's one resource about plants that are sensitive to different air contaminants, but there are many others too: https://www.apsnet.org/publications/plantdisease/backissues/Documents/1982Articles/PlantDisease66n05_429.pdf.