Obviously we mostly don't know what were average Soviet citizen' thoughts on this topic but thanks to digitized diaries from that time we can tell what some of them thought (or, what they thought and wrote because sometimes it'd be dangerous for them to write what they really thought, even in private diaries).

It seems that people didn't know about Beria's arrest before the announcement although this arrest seemed to be inevitable, it was just a question of time.

So, just some quotes for you (with my rough and shortened translations):

Ángel Gutiérrez (he is originally from Spain, but lived in the USSR at the time):

June, 11: I never liked him, but this is much worse that I thought. What a bastard. Scoundrel! He sold his people to the agents of imperialism... He's a killer with animal instincts... It's impossible for to find words to describe how I hate him.

Я никогда не испытывал к этому человеку никаких симпатий, но такой подлости от него никак не мог ожидать. Сволочь продажная! Подлец! Продал свой народ агентам империализма! Жажда власти — вот настоящий Макбет. Звериные инстинкты проснулись в этом убийце. Ещё недавно, 9 марта, он на трибуне Мавзолея, над телом великого Сталина, подлец, произносил такую речь! Нет слов, которые могли передать всё моё возмущение, всю мою ненависть к этому преступнику!

NB: This was written in June, not July. And it's the only diary in that archive that says bad thing about Beria before his arrest in July.

Erlena Lourie (a schoolgirl from Leningrad):

July, 10: Beria is the enemy of the state! It's horrible! What a year...

Берия — враг народа! Ужас какой-то! Ну и год выдался...

Rozam Nazirov:

July, 10: Beria is a bourgeois degenerate, enemy of the state, adventurer and, more than that, an agent of the foreign state... Beria is an enemy. Traitor.

Берия — буржуазный перерожденец, враг народа, авантюрист, даже наймит иностранного капитала. От этих слов звенит в ушах. Берия — враг. Предатель.

Nina Pokrovskaya:

July, 10: Something bad is happening in our country. Newspapers said that Beria is an enemy of the state. That he tried to do his best to destroy our state. To achieve this, he became Stalin's friend and could approach him any time, day or night... There is a lot of gossip about this in Moscow, but nobody knows any real facts yet...

В нашей стране происходят нехорошие дела. Сегодня в газетах сообщение о том, что Берия оказался врагом народа. Всё делал для того, чтобы подорвать наше государство. Для этого добился дружбы со Сталиным и был вхож к нему днём и ночью. Восемнадцать лет он был министром внутренних дел и первым заместителем председателя совета министров. Теперь он снят с обеих должностей. Слухи по Москве ползут самые злые, но никаких подробностей, конкретных фактов против Берии пока никто не знает.
So, yes, people already knew Beria was in trouble, but the level of the secrecy was quite high, so it seems almost nobody knew anything else before this announcement was made.

Or, maybe, no one wanted to write something about that in their diaries.

If you know Russian or ready to use Google Translate, I can highly recommend this resource, all these quotes from real Soviet diaries were found there. It's not in any way really representative (you have to be quite unusual to write a diary and not to lose it during your life and to give it away for digitizing after all), but still it could be very interesting reading if you want to feel a zeitgeist of a Soviet life.