Surely the biggest issue here is that I'm trying to learn usage from anime dialogue, which should almost always be taken with the biggest grain of salt possible. But it comes up often enough that it's at least a trope, if not a common real-world usage, and I haven't been able to figure this out.
Individually, I know what 私
("I") and ね
(often a particle for "right?", or "hey") mean. But when they are combined at the beginning of a sentence, simply combining those meanings seems insufficient to explain what's being communicated.
Usually it comes up when a female character is talking about herself, confiding in a close friend, sounding something like わたしね{LLLH}. In fact, right after writing this, I happened across an example 19 seconds into episode 17 of 四月は君の嘘, when Kaori says 「私ね{あたしね}...」, trailing off right before the theme song starts.
Not sure if more details are needed, but I will certainly make clarifying edits in response to comments, so fire away!