Why don't we get regular checkups for mental health, as in physical health? It's standard practice to get a checkup from your primary care doctor once a year. Why is this not so for a mental health professional? Is your primary supposed to catch possible mental and emotional problems and refer you to a specialist?
The short version is that in Europe and North America, at least, the answer is predominantly yes: your GP is supposed to refer you to a relevant mental practitioner on a case by case basis. This can be prompted either by voluntary disclosure (i.e., the patient brings up mental health issues themselves) or suggested based on comments made by the patients. GPs do not generally have a responsibility to 'catch' mental health problems unless they are potentially life threatening (e.g., suicidal ideation).

This is the most that can be said without knowing which country or countries you are interested in. There are vastly different models of mental health care at work in various countries, and the practice of mental health care varies accordingly. For example, the United States' health care model is primarily based on private insurance, while Britain's is an integrated public-private model, which affects how real practice is conducted. Besides the different incentive structures, countries also have different philosophies of mental health service. 

You can often find information about a particular country's mental health care model and philosophy on the website of their public health institution (e.g., department/ministry of health).