You as an individual are basically powerless. It would need to be some blatant anomalies for you to discover it but it has happened.
One big problem as an observer is you don't get to see the cards they muck. To perform a statistical analysis you really need that. If someone gets AA too much is the easy part. Did they fold JJ because QQ was out requires full access to the data.
Your distinction of honesty of the implementation vs a weak random algorithm is mute. As an external observer all you know is what came out. Unless they disclose the implementation.
There is not a method to determine if a site is rigged. I only know basic statistics. If you had access to the data in 10,000 hands you might be able to show a statistical anomaly. Same guy never got a card under 10 and got AA 1/40. Proving random would take millions of shuffle and even then you would have confidence interval of like 95%. The number of possible shuffles is like 50 digits long.
There are testing agencies like ecorgra that sites may submit to.
People blame bad beats on unfair shuffles. Statistically bad beats happen. Runner runner (need the last 2 queens) is 1/756. It happens and people blame it on unfair. If it happens 1/30 for a player in 10,000 deals then that is flag. But if it happens 1/400 would take billions and billions of deals to prove a statistical anomaly.
A perfect shuffle is mathematically defined. I produces all permutations equally. The Fisher Yates shuffle has been around since 1938 and yet at least one site managed to get it wrong. The common mistake is to produce too many permutations. Another is to miss shuffling the last card. So if you watch enough hands and a card never comes up you know it is never going to come up. If all decks start with the same last card then a big hole.