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The best way to gain exposure on Twitter is to be active and relevant towards a target audience. If you nurture a reputation as an expert in your field, then you are giving potential employers a reason to notice you.
Participate in talks and discussions, follow the latest trends, and offer your perspective to other industry players.
Whether it is a PR representative, a CEO or a top player in your target industry, they have to deal with sycophantic communication every day. If you contact one of them directly with an express intention to plead for work, a positive response is unlikely.
Be friendly, appear interested, and don't push yourself forward -- simply by being connected through social networking opportunities will find themselves in your news feed. Be strategic in who you follow, re-tweet updates that interest you, and contribute to discussions.
There is no reason why online networking has to remain strictly online. If you follow events, conferences and talks relating to your field, it is possible that you may meet those you are already in contact with -- taking your job search a step further.