How accurate are allergy skin prick tests? are 'true' allergies only manifested by nasal/eye secretions?

Quite accurate: if performed correctly with controls by experienced staff, you don't need to have eye / nose symptoms if you have allergies, asthma is an allergic disease too, some patients do have significant reactions on prick testing yet they have no manifest allergic disease,

Not just eyes/nose: True IgE mediated allergy can affect all external body surfaces including eyes, nose, intestinal tract, skin, and lungs. As for skin prick test, the accuracy often depends on the extracts used (standardized ones are better) and the size of the reaction (the larger the more likely). Although a positive response indicates sensitization, the test results need be interpreted by an allergist.