When I first started my nose ran constantly. It took several months for it to stop. I don't know what people thought, and didn't really care as I knew what was causing it -- as JNK stated it was getting used to therapy.
I developed substantial nasal congestion issues beginning the morning after my first night with new CPAP. I do have seasonal allergy issues this time of the year, but this seems kind of coincidental. I use non-heated humidification and a nasal pillow mask. My wife tells me I don't open my mouth during the night. I'm going to try without humidification for a few nights to see if that helps.