can equity analyst trade their own money

On the sell-side, you're heavily restricted on how you can trade. Typically, you can't trade within your own coverage universe (firm policy) or even the vertical your stocks are in (e.g., Tech, Healthcare, etc.). Outside of that, you can trade most securities that aren't on the firm's restricted list. Also, you can't sell a position for 30 days unless you're in a dire financial situation.

To sum it up, it's not worth it unless you're building long-term positions in company's outside of the sector you work in.

As a buy side analyst, I had to ask permission for the firms compliance for every trade (buy and sell) and I had to hold my positions for at least 30 days.

It's a pain. I just stopped caring about my PA entirely. In addition to everything Shawn said, you have to wait for clearance from compliance first before you buy anything (doesn't usually take long but still sucks).

One exception at my bank are ETFs and/or mutual funds. ETFs and/or mutual funds do not have any restrictions here.

that really sounds like pain in the ass. but if yo have positions before joining the firm, and your positions are in their restriction list, do you have to sell them before joining? I assume options are also restricted correct?

Interesting about ETF/mutual funds. I suppose mutual funds are a good place to park your money and just forget about it, but with ETFS you can actually act on some trading ideas/strategies