1) Observe their behavior. Look at the way they act. How do they treat others? Are they inconsiderate and/or rude? Are they quick to join in on gossip? Are they quick to judge people? Do they continuously concern themselves with what other people are doing? There's a difference between someone who wants to be helpful and someone who's just being nosy and wants something to gossip about. However, that alone is not enough to determine whether someone is trustworthy. Someone that seems to be involved in a lot of drama is someone that you need to be cautious of - there is a reason why they don't get along with so many people.
2) Listen to them, communication is important. When they talk to you, do they redirect their conversation towards other people - and if they do, are they saying negative things about them? If this person is telling you things about other people's private lives for the sake of having an "interesting conversation", you should take that into consideration when determining their trustworthiness. It doesn't matter if they give out their name or not; someone who talks about the business of others (with strange intentions) after they have been personally confided in is someone that you should be wary of and you should begin to question whether you can trust them or not. If they feel comfortable with talking to you about their own personal business (e.g. things that are bothering them, asking for guidance/advice etc.), it could be a sign that they somewhat trust you. Perhaps you should take a chance and trust them too.
3) Reliability, can they keep promises? Everyone lets someone down from time to time, but if someone is continuously letting you (or anyone else) down at the most important times with little-to-no excuses, then they aren't very reliable. Part of having trust in someone derives from being able to have faith in them and rely on them no matter what your situation may be.
4) Honesty. Someone who constantly lies is probably doing so as a result of trying to save themselves. Once again, intentions are another thing to consider, but someone who constantly lies about themselves in order to look good may have something to hide. If they can lie about their own life, then imagine what else they are willing to lie about. Someone who has to lie to get themselves out of a negative situation isn't trustworthy, just like someone who maliciously spreads lies about other people is spiteful.
5) Personality. Other things to consider are whether they are a self-centered person or not. Self-centered people only care about themselves and are more likely to let you down on more than one occasion than anyone else. Some people will make exceptions for the very few people that they care about, but if they constantly make themselves top priority then those people have the potential to betray you when given the chance.