Let's denote the number of people who speak both the local language and pidgin English as x. According to the principle of inclusion-exclusion, we have:

\(350 + 200 - x = 500\)

Solving for x, we get:

\(x = 350 + 200 - 500 = 50\)

So, there are 50 people who speak both the local language and pidgin English. The percentage of the population who speak both languages is:

\(\frac{50}{500} \times 100\% = 10\%\)

So, **10%** of the population speak both the local language and pidgin English.