There are not many large cities and if you use some common sense and street smarts, you are unlikely to run into any trouble. The police are known to be corrupt, and if you are pulled over for any reason, you will almost certainly be expected to pay a bribe. In Asunción most cops are not corrupt. In the cities crime is common, though not as rampant as in other cities such as Rio de Janeiro, São Paulo and Buenos Aires.
Ciudad del Este is a hotspot for illicit activities, such as money laundering and counterfeiting, but that should not affect your travels. That said, you will want to keep an eye on your bags and wallet here, as you would do in any other large city. Generally, as long as you aren't involved in drug smuggling (inadvertently or otherwise), and are alert to pickpockets, you should be safe most of the time.