I am currently running a very simple side project which I hope will lead to something bigger.
I see quite a few tweets from my competitors customers saying they're using their product for a particular reason. However, I believe I am offering something a lot more valuable but I can't just reply to that person and say "hey check us out", can I?
It feels weird to me although I honestly think they will visit my site and possibly use it.
What are your thoughts on this?
This sounds like spam to me. If they are tweeting about how happy they are with your competitor's product, they have no reason to switch, especially if it is a paid product. Unless the tweets indicate that they are unhappy with your competitor's product and are looking for an alternative, I don't see this gaining you any customers. In fact, it will likely have the opposite effect. So no, I wouldn't do this. And if Twitter gets enough complaints about your account, they may disable it.
You can, however, use the tweets to gain valuable information. You can learn information like what features they like the most, what features are missing that they would like to see, and how they are using the product. You can then incorporate what you've learned into your own product to make it more appealing to people in the market for a product like yours.
Additionally, you can use this information to write blog posts on your product's website to attract people that are searching for that solution in Google.
An alternate technique I've heard of people using is to bid on [competitor name] keyword in AdWords. I've never done this myself, but I know it's done. This may be a sleezy thing to do, but at least it's not spammy.
Edit: See this post that shows how terribly wrong contacting your competitor's customers via Twitter can go.
If the product is paid and/or subscription based, I would think the probability of "stealing" them is close to zero.
However, expanding on Zulys answer, one can learn a lot from them. Reaching out to them to offer some useful actionable research AND learning what made them pull the trigger (subscribe) + what they consider important would be much more valuable.
I get it all the time, companies do that a lot.
Honestly, it looks a lot like spam, so I don't usually click it. There's nothing that prevents you from doing this, I'm just not sure how effective it's going to be. Especially if you're going to do that manually.