“How is a sinus infection diagnosed?”

Sinus infection is diagnosed by taking a sample through the nostrils and sending it to the laboratory for culture. Usually the ENT (Ear, Throat and Nose) doctor would be more precise in taking those samples. They also have camera that can go further up to check for any focal accumulation of infection/inflammation. Thank you.

CT sinus or a nasal endoscopy by an ENT.
Clinically a bacterial sinus infection is diagnosed with symptoms of fever, facial pressure, colored nasal mucous, congestion that does not improve for 7 days. If you are having improvement, its typically viral and does not need to be treated with antibiotics.
Either way, you can simply treat yourself with OTC nasal steroid spray daily for 2 weeks, 3 days of nasal decongestant spray for 3 days only, and sinus irrigation. You may like to add a menthol additive to your irrigation, Akylol for comfort.

A sinus infection is typically diagnosed based on history and physical exam findings. If the history and physical exam findings are not classic for a sinus infection or another diagnosis is being considered, a CT scan of the sinuses may be considered. Often times a trial of antibiotics we'll first be performed to see if this resolves the situation. However, this depends on a detailed history and physical examination to see if this is warranted.

Mainly by medical history. Physical exam may reveal supporting findings. A large percentage of "sinus infections" are not. Rather intranasal conditions, headaches, viral infections, etc. Best to have nasal endoscopy and a CT scan. If recurring or frequent check risk factors, i.e. smoke, dust, allergies, air contaminants, etc.