Catalonia could be a good case study: it has 4 police forces, but I wouldn't call it competition.

Each serves a different master:

👮‍♂️ Mossos d'Esquadra → Catalan gov

👮‍♂️ Policía Nacional → Madrid

👮‍♂️ Guardia Civil → Spanish Interior

👮‍♂️ Local cops → Mayors

Result? Not checks & balances, but tensions in the streets between the different forfes that end up harming its citizens (2017 referendum, 2019 protests).