I have a personal Twitter account (http://twitter.com/alleverybody), which is very rarely related to my professional life. Although I'll ocassionally tweet about some sort of workplace challenge, fail, success, etc., my snippets are 140 character reflections of ideas and experience.
I handle professional client-side twitter accounts as well and I often post a range of tweets via my phone, whether personal or professional.
And this requires careful execution as I never want my personal tweets to end up in the wrong pipeline. This happens to professionals all the time and often with disastrous results. That said, I never personally tweet angry or vindictive rants, I don't name names if I have a work-related conflict and I never slam people who I know personally. Why? Because my Twitter account is transparent. While I use the nom de plume "Alleverybody," my name is right there on the account. I also post my photo. So I'm pretty easy to find.
My Twitter account is also sacred and if I were to go off on a twittering rant about a troublesome client, there's no doubt I would be discovered. So I simply don't do stuff like that.
On the flipside, I never correspond with clients or co-workers about work matters on my Twitter - that's not what I have it for (by the same token, I no longer open my work emails on weekends - the truth is, most stuff can wait until Monday). Once you start answering and abiding by weekend work requests via your Twitter account - you're doomed.
By doing this, you're setting a precedence that devolves your personal Twitter into one filled with inconsiderate demands, unreasonable requests and useless anxiety.
I had this happen on a recent Saturday morning. Someone decided to tap into my personal twitter with a work-related request that was not only time-consuming but unecessary (except to fill her agenda). So rather than go down that road and acquiesce to this person's request, I tweeted (within nanoseconds of her request as I knew that she'd see it because she follows me) this in response to seeing a person walking into a neighborhood Starbucks rocking a major fashion faux pas: "In case you're tempted - don't ever wear horizontal striped pants..."
And I went on to enjoy my weekend.