Complex sentence: Thief-takers met the demand for policing that public authority boosted, publicised their skills and took advantage of the situation to create their respectability among citizens; this conduct also provided thief-takers with a safe front for illegal trafficking.
Split into simpler sentences: Thief-takers met the demand for policing that public authority boosted; they also made their skills known at any chance and took advantage of their public role to gain respectability among citizens. This overt conduct also provided thief-takers with a safe front for illegal trafficking.