<< How can one tell if they are possessed or otherwise influenced by other spirits? >>
How does the host know? Sometimes they don't until the exorcism. Subjects have reported that they knew they had somehow submitted their entire will/soul to some foreign influence but they weren't sure just what it was - only that it really wasn't themself.
Subjects sometimes have periods of lucidity where they seek help, but most of the time family and close friends have to find the help for them. (Then there are the truly sad folks who have zealous relatives who ACCUSE them of being possessed, but the accusers rarely know what is what).
An exorcist or a demonologist determining whether someone is possessed requires ruling out mental or physiological illnesses or just strange behavior. Generally there is an investigation and interview to ascertain if certain criteria are present. This testing of criteria continues in the preliminary stages of an exorcism as well.
There are some folks, including some deliverance ministry types, who think ANY negative or destructive influence is a demon. I do not share this view. Alcoholism, for example, is not a demon. The simple test is that alcoholism is impotent without alcohol. A demon can do that destruction to someone without the bottle of bad whiskey.
Possession involves the gradual but extreme replacement of a person. If you ever run across it you'll know exactly what I mean. Lights on, somebody's home, but it isn't the owner.