Elementary sentences containing the quantificational determiner some seem to be ambiguous between a ‘weak’ existential meaning 9 and a ‘strengthened’ some but not all meaning 9+. The strengthened meaning is commonly assumed to be the output of a general enrichment mechanism, call it G (for ‘global’), that applies to the weak meaning of the sentence: G(9) = 9+. The application of G has been shown to come with a processing cost (e.g., Bott and Noveck 2004). We used a self-paced reading task together with offline comprehension questions to investigate the interpretation of sentences containing some when embedded inside a disjunction, a position that G cannot access. Our findings suggest (i) that the strengthened meaning 9+ is available in embedded positions, suggesting that a mechanism of local strengthening L must be available: L(9) = 9+, (ii) that local enrichment can be facilitated by global pragmatic pressures (Chierchia et al. 2008, Mayr and Romoli 2014), (iii) that subjects can be quickly trained to systematically prefer one of G or L to the other, (iv) that application of L, like the application of G, comes with a processing cost. We highlight consequences of our findings for debates about the characterization of enrichment mechanisms, focussing on the relation between G and L.