If you've been anywhere near a social platform, you've seen messages like this: "Friend A likes this update" or "Friend B" has read this article" (and, clicking on the article title seems to always require you to hand all your information to the publisher before you can see the content.)
Facebook, in particular is very keen on you seeing your friends' activity. LinkedIn, too, wants you to know that a connection has followed a company, connected with someone else, answered a question, joined a group.
The problem with that is exactly the same problem as talking to a friend who walks you through every step of their day, without degree or distinction; waking up, brushing teeth, eating breakfast, commute, meetings at work, home, etc... no one wants every single detail of your day. What we'd like to hear is the juicy bits, the good stuff. ^_^
Each social platform is a different conversation. Even when you have the same people talking about the same thing in the same place, there's subtle differences in audience, tone, attention, and acceptable length. Posting Twitter conversations on LI is meaningless to anyone not already in that conversation.
Be mindful of the difference between "activity" and "content."
Facebook wants us to know all our friends' activities. What they read, what they commented on, what games they played, etc.
If your Twitter feed is automatically sent to LinkedIn, all your contacts are getting all of your activity, your RTs, your @s, your replies. The only thing that will be meaningful to many of those people in all of that is any content you share. And how patient do you think your contacts will be when they have to wade through 3 dozen chatty posts to get to a brilliant piece of content?
Quora, a platform I adore, yesterday launched a new feature that shared your Quora activity on your Facebook Timeline. I enabled it and ten minutes later, disabled it. I have no doubt my friends on FB would be interested in some of the answers I post to questions there, but equally, I have no doubt at all that no one not already on Quora cares which other answers I upvote. While the content I generate there may be of interest, my activity on Quora is entirely irrelevant to the folks on FB.
Small and medium sized businesses on Twitter, and individuals whose expertise is their business, often have only one Twitter account. To be authentic and real, these people tend to chat as well as share good content from this same account. This works really well on Twitter, where each tweet is viewed individually but, when it becomes a stream of half the conversation on a Twitter feed embedded on a web page, it simply makes no sense at all. Like sitting next to a person on the train who talks loudly enough that you can't not listen, you're getting an intrusive half a conversation you don't really want to hear.
Worse, when companies keep their Twitter account for purely professional contacts, that embedded stream becomes an obsessively narcissistic stream of "me me me." Again - it works fine on Twitter, but watch where else it gets shared or you can seem like you are incapable of listening to others.
So, by all means, share content! Just be mindful that your not drowning it out with all of your activity.