The only way to make Twitter useful for your ministry is for your ministry to have its own Twitter page and for someone to take ownership of that page and regularly post tweets and use the other Twitter tools to help create a presence for the ministry and draw people who use Twitter to your website resources or other resources you think valuable.
Twitter is like a bulletin board where you post short messages that people can just touch and be transported to whatever you want them to know about. Cool. On the receiving side, if you have signed up to “follow” someone’s, or many people’s, tweets (posted messages of 140 characters or less), it’s like being at a large convention and walking around listening to snippets from various conversations as you go. This is a very useful way to stay tuned in to people and topics you’re interested in, like what’s happening at your ministry or with people associated with your ministry. Very cool. Your Twitter page will only be worthwhile if someone takes responsibility for using it to the church’s social media advantage. For this you need a social media strategy and plan and stuff.
Twitter is generally considered to be one of 4 or 5 key tools for organizations that have a social media strategy and are implementing it. If you just put a link to Twitter on your home page, with a cute little Twitter bird symbol, it will only be useful for people who already have their own Twitter account and want to add a link to your ministry to a Tweet they post themselves on their own Twitter page. Hardly anyone would ever need to do this, since you can add a link to anything into any Twitter post you create very easily.
First, you need to learn enough about what Twitter and other social media tools can do if used well. Then, you need to write down a strategy for how to use Twitter to help your mission. Then, find someone, or get yourself ready, to take charge of your Twitter strategy. Twitter champions really have fun with it. You can, too.