I know that 頑張らないと would mean "I have to work hard" (as if I am forced to work hard). What if I see someone (my friend) working hard and I am motivated to work hard too. In this case, would 頑張らないと or 頑張らなくちゃ be ok? Or is there a better phrase? 頑張らなくちゃ sounds very forced to me.
I guess in the context you describe
would work fine, because it's understood that you are "forcing" yourself to work, which is just one way to express your own motivation.
But you could just as well use
(Here 私 is just a place holder for a context-appropriate personal pronoun, for example 僕 or こっち could also work.)
Your examples are the shortened sentences, Japanese language is fond to omit the part which is not necessary, unlike French or Spanish needs to say all.
The original sentences are:
あなたは頑張らないといけません ≒ あなたはそれを頑張ってしないといけません
あなたは頑張らなくてはいけません ≒ あなたはそれを頑張ってしなければいけません
Which provides almost no difference at all to me. Only the 頑張らなくちゃ sounds a lot softer or a feminine way.
And 〜しないと has a better way to translate in English, 'Do or not'.
'Dodge or hit your head!' 避けないと頭を打つぞ!