The latter. I've never seen either a bloodstain in either form and I died many times in my session in areas I could easily return to and would know where to expect to see it. I don't expect (real or faked) online-only features lol that'd be such a scope bloat for poor Lilith!
Interesting, I wonder if that's a glitch in your game somehow? I just got myself killed by the fella near the great bonfire lamp and sure enough my echoes went into one of the enemies down on the street. I only saw one loose bloodstain but I've seen glowy eyes several times. That's very mysterious! I don't have any solution unfortunately but rest assured it is supposed to be there, maybe it's something to do with the area transitions? I hope it starts working for you, you deserve your blechos! (And hahaha yah can you imagine how much work online would be? Plus totally not ps1)