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I am wondering whether it is polite or typical for an Italian person to say 'bye' or 'ciao' at the end of a phone call or when they leave the house.

I know an Italian who never signs-off on a phone call, they just hang-up and they also don't say good-bye when they leave.

Is this typical or is this person just impolite?

It's pretty unusual and impolite, and also shows an ignorance of at least one of the functions of language.

As for which way to close a conversations, friends or close colleagues will use a normal Ciao (there is a recent custom of closing a phone call with a sequence of generally three ciaos). People who are in less intimate relation would say Arrivederci (even if A risentirci would be more correct), A domani (“till tomorrow”, if that's the case) etc.

Bye or words in other languages can happen, but it would be an unusual and joking way of speaking.

More or less all of the above (but for the thrice-repeated ciao) holds for meeting in person, for when you leave a shop etc.

It is not normal to sbattere/chiudere/riattaccare il telefono in faccia (hanging up a phone call) nor andarsene alla chetichella/all'inglese (slipping away quietly without saying goodbye).
I personally find strange many TV series where they never say goodbye at the phone, and most of them are American.

Although I must say that in some circumstances it could be normal if they are work related calls and the job implies making a lot of these calls especially between the same parties so that saying hello+goodbye every 5 minutes for a call would actually take away productive time.

Usually is typical for an Italian person to say goodbye with "ciao" or "ci sentiamo (dopo, domani,...)" at the end of a phone call or with "arrivederci" or "ci vediamo" when leaving the house; it also depends whether it is a formal situation or not.

Obviously it is impolite to hang-up without saying goodbye or some kind of salutation, but that's true no matter what nationality you are.

Moreover, regarding some formal ways of signing off a phone call you can find in this blog some examples:

Telefonare in italiano: come terminare una conversazione? (How to end a phone call?)

1. 
   Signor Bianchi, è stato un piacere parlare con lei. – Mr. Bianchi,
   it was nice talking with you!;


2. Me ne occuperò domani. – I’ll do that tomorrow.


3. Lo verificherò subito. – I’ll check it right away.


4. Le invierò il preventivo per posta. – I’ll send you an estimate by
   mail.


5. Le invierò i dettagli via e-mail. – I’ll send you all details by
   mail.


6. Sono in attesa di una sua conferma … – I’m waiting for your
   confirmation...


7. La ringrazio per l’aiuto! Arrivederla! – Thank you very much for
   your help! Goodbye!


8. 
   Arrivederla signor Bianchi, a presto! – Speak to you soon, Mr.
   Bianchi!