I contacted a professor at the local university (I live near a top uni in Canada) who happens to study something that I would be interested in studying and they were very receptive to taking me on as a student!

However, we have been exchanging emails and I'm getting worried. They replied very quickly to my first emails, but said they were in a conference out of town so we didn't meet in person until around a month after our initial correspondence. Then after we met up, I sent an email the next day following up on some things we talked about during our meeting and they didn't reply for another month... I sent maybe 5 follow up emails spread out across a few days or weeks to try and get in contact with them. Finally, they responded saying they were busy teaching a class - which I completely understand, professors being busy with their teaching and research and considering I'm not even their official student. We set up another time to meet up in person, and during the meeting we set some goals before the application deadline. They said they would supervise me to write 2 separate pieces in preparation for my Masters / PhD. I sent another email the next day following up on that meeting and outlining the two pieces. This was a month ago, and despite a couple follow up emails, I haven't heard back.

Whenever we meet up in person, they are very receptive and helpful. But once I leave their office, it feels like I can't connect with them at all. I'm worried that if I do begin a PhD with this professor that these communication difficulties may remain... Am I overthinking this? I know that professors get busy, and considering I'm not officially a student yet, this professor doesn't "owe" me anything. I have been working for the past 3 years and I'm used to a work environment with a lot of communication. Are my expectations inaccurate for the postgrad world?

Hello. From my own experience and experience of aquantances who apply for PhD, it is not unusual. Any reply you get from them, might be the last communication and he might not reply again forever. Scary? yes but unfortunately true. This is typical. If you are their student, this behaviour is going to change a little. At least you can come to their office. But now is the real problem to convince them to supervise you "in real actions" not just nice words. Of course funding plays the dominant role here. They might see you as a good candidate but not good enough to spend their fund, if they have any.
In conclusion, just wait. Continue sending follow up mails but without getting hopes high. Try with other professors in same or another university.
I am usually optimistic but I do not want to give you a false hope. This is how academia is. Finding someone who is interested to supervise you is good. It means you have a good CV and a good chance to advance in your career either getting a PhD or work somewhere else. But getting a real paid PhD position is completely different thing.
I wish you all the best. Just try and follow up and do not give up.

That sounds normal for academics. His initial enthusiasm sounds very positive as the long correspondence delays is normal. If anything the initial speed sounds like a green flag.

I am just curious if you have talked about funding. Finding funding is the most difficult part of organising a PhD and I would talk with him about your options. It may be that he is waiting on a funding decision or waiting for another round of funding to start. I think you should be worrying about this and your proposal more than correspondence speed. Though goodluck!

Your last two sentences sum up the problem here.
That sounds like a huge number of emails for you to be sending.
A PhD is completely different from industry and you would be expected to be more independent.
To give an example, I met with my supervisor once every 6 to 8 weeks for about 3 hours with only a single email summarising my progress and to book him for a chat. He didn't expect to hear from me anymore than that and if it had been up to me I'd have kept our meetings to once every 6 months and only when I needed it.

I don't think I ever got a response to an email unless I was setting up a meeting or asking him a rare direct question.

Thanks for all the replies and the reassurance - I mainly wanted to get my expectations in check! It's good to know this is the norm.

One thing to note though is that during the meetings, they mention that they will email me certain pieces of information - this is partially why I email immediately after the meetings. I could understand if they get busy and forget... but is it unreasonable for me to send follow up emails inquiring about things that they had said they would email to me?

Again, sad but still normal. Promises made by academics do not count. As said by rewt, funding is the key. No funding means no PhD. Is your topic good enough for them to sepend their funding? Be carefull when professors have "real" funding, they sometimes change their minds and decide to advertise for a PhD to choose from many candidates rather than accpeting one. THey say many things but when it comes to money, everything can change.