I tend to write better when I'm writing in first person. However, one major barrier is when I'm writing about a character who has to "hide something" from the world. Since I'm writing from his/her perspective, the character tends to be an open book. A person's actions or dialogue can't be as ambiguous as I might wish them to be. Is this unavoidable?
Writing within the first person seems to be quite a difficulty to do for many writers, especially when the piece you're writing is rather long. If so, and you're up to the challenge, having the character not becoming an open book is avoidable. You just have to remember that, even though you're writing from within the eyes/mind of that character, he/she doesn't have to necessarily say every little thing they're thinking/observing. It's a lot like third person limited. Example, you have a high school guy that feels pretty lousy every time this girl walks by. You describe the built-up rage and the weight of the insane depression coming down upon You. You describe the casual steps of her shoes and the fact that she never turns your way. The reason is because the girl dumped you for another guy (perhaps your friend?). But you don't have to tell the reader that until you feel the need to.
The book Speak did a good job if this, The whole book the main character is kind of treated as an outcast by her friends, she stops caring about what she looks like, she has a sort of major crisis of self, and she can't talk about the reason why to anyone, so the reader doesn't find out to the end that it was because Major Spoiler in white she was raped and that's secret she's been hiding the whole time So yeah this can definitely be done, you can tell about the symptoms or consequences of the problem without giving away what the problem/secret is and this also adds suspense.
Quite a lot of my writing is in first person for some reason. What I would say about not giving away too much is that you need to be careful not to give too little. It seems an obvious point but you can easily have your reader turning to something else because he or she can see the obvious ommision of details. Have you ever found yourself switching off whilst watching one of those ridiculous rom-coms where the main guy just needs to say one completely obvious thing to save the day. It's someething that you know you'd have said. It's painful to watch and to read. I think you should always try to work out who your character is before you get into the meat of the piece. Once you're in character, just be natural and trust your ability to take it into the second draft.
I won't go so far as to say it's unavoidable, but it's a characteristic of the first person POV. First person is an introspective viewpoint, and it's unnatural to conceal important information from yourself. You could tell a story from a position of delusion, but that in itself is a choice that has broader implications than simply concealing key information for dramatic purposes.
Have another character keep the secret. That way you can still keep the secretive element and write in a POV that you are comfortable with.
first person Writing in first person is a really fun thing to do, if you can pull it off. You can avoid telling the whole story, like those mentioned above me, because lots of times, if you think of a real life example, we can hide things from ourselves, too.
Third person gives you a chance to become the psychologist. You can examine the characters and even start defining why they act the way that they do. First person, however, gives you the mind of the protagonist. Noe1111 is completely right, we are fantastic liars when it comes to ourselves.
I kind of like the idea of having a narrator who doesn't always tell you everything, or maybe even deliberately tries to mislead the reader. Not on big things, mind you, just on little details or things that aren't important at the time. For instance, let's say he starts out hating one particular character, but gradually starts to like her. But he doesn't want to admit it. When the situation calls for it, he could outright say that he still hates her, while his actions indicate otherwise. Maybe someone catches him staring at her with a little smile on his face. The same thing once happened to me in the real world, so why not in a fake world? Might be a hard thing to do well, and I'm sure it could get confusing if done poorly. I've never tried it, but I intend to.
I kind of like how Catcher in the Rye did things. The protagonist was a dishonest person by nature so you had to question everything he told you. Not 100% sure about that since I didn't like the book, but it made you think.
Also, you don't need to position the reader in the head of the narrator. You can fairly easily stop your reader getting into the head of your character in first person if you think about things a bit.
My current story is in 1st person. I found it to be the only thing that really worked, because the MC's inner life is half the plot. When things need to be omitted for dramatic purposes, I simply shift his focus outwards onto the world. I make his senses tuned onto things that are happening right here and now before him, so that he (and the reader) have no time to focus on inner dialogue. Don't you know this from your own life? If you're late for the bus or something, you don't really have time to think about inner stuff. Brain's occupied with "Where's my bus card?" "Damned shoes won't tie...!" etc. And then there's the whole thing about keeping secrets from yourself. Everyone does it, all the time! No one is less objective about your life situation than yourself.