This, as with nearly everything in bridge, depends on partnership agreements and style.
However, taken in the standard methods, 2c is the only real forcing bid available. As such, it usually promises at least a limit raise, although there are other scenarios where there's no better bid; but most commonly it's a limit or better. So, partner has at least a limit raise, and could have more.
After the 3c, pass is not a forcing bid in my opinion. Pass is a "I have nothing interesting to say". You don't have a six card spade suit, you don't have anything unusual. The real question is whether you have a stronger hand than expected for an overcall, and that is definitely not the case in standard - standard would overcall all the way to about 16-17 (before you're in the double-and-bid range). So, you have a normally shaped hand (roughly), 5-2-2-3 (well, I hope you have a 13th card there somewhere...) - so pass is perfectly fine. That lets partner make the choice - what did that limit raise mean? Let you know. Pass is technically an option for them, though usually you'd want to compete.
Now you're not making 4S (losing one in each suit, if they just take their aces after getting back in from a club lead). If you'd overcall with that, then you need to differentiate that from what you actually have here - because your actual hand could play game opposite some limit raise hands, while the example hand here definitely doesn't want to. But if you play more tight overcalls - like, six losers or less or something like that - then you probably don't need to. Me, I would at least think about overcalling with that hand but probably wouldn't actually do it (the Jh is worthless, and the Jc almost worthless, so I'm not sure I rate this 8, and the broken spades and lack of aces is worse). Make the Ks an Ace, and it's more palatable - but we also then probably make this game.
For your bid, the key is to have a good understanding of what all of partner's options are. What range will partner pass with? Will they ever pass, or do you prefer to compete with any hand that would limit raise? What does X mean - is it penalty? What does X mean for you (you do have 3 clubs and a hand that plays decently on defense, and West probably has little to nothing)?
If Partner would compete in basically any auction, and never pass, then 3S is not telling you much - and P should be doing something else. I would argue that there should be some hands you would pass there, mostly because otherwise there's no reason for E not to bid 3c there and eat up space (and you lose a useful bid). But it'll be rare. And either way, P's hand is not one of those pass hands. So let's evaluate their 3S.
Most likely, Partner should be showing something here other than 3S, given they probably do have a slightly better than limit raise hand. They have a singleton diamond, which plays well no matter what you have there (and you're rated to have some). They have KQ hearts, which is nice (even if it is likely to only be worth 1 trick). They have the ace of clubs, so E can't easily get much there. And, most importantly, they have four spades - so there is 9 spades on your side!
I think you could make an argument for 3H, to show that absurd length in hearts, mostly because the hand is so major-heavy and shapely that there's concern you might miss a weird slam. Their hand is a three loser hand outside of spades - give you
All of a sudden it's an odds-on finesse from six hearts! So not something North should be ignoring. If they bid 3H, you either bid 3S (minimum) or 4S (not minimum) if you are uninterested in hearts or slam. If they see any other bid from you, it's a specifically meaningful bid - agree on what, but probably controls of some sort, indicating you are interested in the weird slam.
All together, I think my answer is you're both somewhat at fault here - maybe. If this is anywhere close to your minimum overcall, then it's less your fault; if it's well above that, then you need to have a bid to make after 3C that shows this hand.
But partner definitely has a good hand - probably better than yours in some ways - and underbid here, assuming you weren't showing an 8 count with your pass. Even an eight count plays okay if it's the right 8 count - so I'd like to see something more interesting than 3S, on balance.