Neither is necessarily the case. Moreover, even if the client is doing absolutely everything locally, it doesn't mean I as the user have any idea how it works under the hood. There's just as much trust involved either way.
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You are able to choose which client to run, or even write your own. If there's a bug you can switch to another one until it's fixed. Depending on your circumstances you can choose between convenience and security. Relays are shared by everyone, and simplicity improves stability.