It is impossible to learn spoken Tibetan without learning directly from a native speaker (just as it is impossible or nearly impossible to learn spoken any language from anyone but a near native speaker or better). However most Tibetans will not agree to help you learn spoken Tibetan. Some lamas will of course.
It is easy to learn Tibetan for the purpose of reading prayers and sections of text. Most people need to devote 2 hrs or so min. a day to do so. You can begin to learn Tibetan for that purpose by going through Cathy Kielsmeier's texts. Then you can begin going through Wilson's "Translating Buddhism From Tibetan" (Classical) or "Manual Of Standard Tibetan" (Modern).
However everyone emphasizes the wide variety of practically mutually unintelligible Tibetan dialects. As I am only learning Classical Tibetan (and really only getting started after years of slowly starting up) I cannot speak to this from experience. English and esp. German, two languages I am fluent in, also have mutually unintelligible dialects (English not so much but one such dialect does exist for sure [I found out that my mother and some of her relatives spoke it] and I personally have extreme difficulty understanding people with typical Bostonian area accents - also most people who haven't lived in Hawaii cannot understand Hawaiian Pidgin which is a real dialect). Linguistically dialect means nothing really other than a marker for variation. Minor variations aren't enough to constitute a separate language. Extreme variations are but people usually just group such extreme variations under a specific language, often for political reasons. However even Tibetans are said to have difficulty understanding one another.