“The Chofetz Chayim used to say that when you feel heavy-hearted, you should speak to the Alm-ghty just as a child speaks to his [loving] father. One does not need formal prayers for this. Rather, speak to your Heavenly Father in any language you wish.” 
Personal prayer in your own language is often associated with Rebbe Nachman’s teaching about the practice of “Hitbodedut” — going off into a secluded area, such as a forest or field, and pouring out your heart to G-d in your own language and your own words.
The Chofetz Chayim doesn’t overtly suggest going into a secluded area, nor does he suggest speaking to G-d this way as a regular, consistent practice (as does Rebbe Nachman).
But he, like Rebbe Nachman, says that you can certainly use your own language, meaning that you can use your own words, too. In Yiddish, that might even mean addressing G-d as “Tateh” (טאטע/father). In English, “father” might sound rather formal. We’re more likely to call him “Dad” or even “Daddy.” We can address G-d this way, too, if it helps us (I’m not sure I’m ready to address G-d as “Pop”). We could also address G-d as “My dear Friend,” or any other personalized expression, depending on our own inner feelings.
The Chofetz Chayim is also saying that formal (i.e. liturgical) prayer isn’t required to speak to G-d, especially when the need is very strongly felt. Prayers are “formal” when the texts have been set and can’t be changed (although there are some variations according to custom and locale). Formal prayers are usually said in the synagogue; some can’t even be said without a minyan (traditionally 10 Jewish male adults; currently, some branches of Judaism accept female adults as well). If liturgical prayer isn’t being used, where do you get the words? From your own heart. Tell G-d what you’re thinking and feeling. G-d already knows, of course, just as Sarah’s unspoken thoughts were known.  But by articulating them, we help clarify them to ourselves, thereby adding to our relief and allowing G-d’s response to be that much more related to our special concerns.
I believe that the ultimate intention of the authors of liturgical prayer was that it should become just this kind of personal expression.
What gets in the way?
Our overall lack of familiarity with much of the liturgy, for one thing. Many of us, who might attend synagogue on Shabbat or holidays, don’t ordinarily attend on an ordinary weekday. For that reason, we lose the chance for a lot of repetition. Repetition, like practicing a musical instrument, can lead us to a certain comfort and flow in saying the words. We’re not always remaining at the level of having to pay conscious attention to the words themselves.
When we’re learning to type, we have to start by developing a conscious knowledge of where each letter/key is. With enough repetition, we internalize the locations and the necessary finger-movements. Then, we don’t have to think so much about each stroke; we can type faster and more easily.
For another example, an actor/actress is given a script — a prepared text — to memorize, in order to know what they need to say in a play or movie. But they’re ultimately expected to do more than simply pronounce the lines mechanically. They’re expected to invest them with feeling and meaning. In the extreme, there should actually be no distinction between the actor/actress and the feelings of the character they’re portraying.
I think that the authors of our liturgy had those goals in mind.
Of course, the Talmud  says that the essential requirement of prayer is to know that we’re standing before G-d (“The King of Kings, the Holy Blessed One”). This “knowing” separates merely saying the words from a contemplative awareness of where, why and to Whom we’re saying them. Kabbalah expands on this by prescribing thoughts about the Seder Hishtalshelut (the emanation of all from the Infinite Divine) into our prayers.
But it can take many years to become that fluent in the liturgical Hebrew prayers. The Chofetz Chayim is saying that we can have that awareness whenever we need it. In this regard, Rebbe Nachman and others seem also to be saying that by talking to G-d regularly in our own words and our own language, we can cultivate greater and greater awareness of being in G-d’s Presence.
“By pouring out our hearts to the Alm-ghty, we become closer to Him and [our] suffering becomes a tool for our [spiritual and emotional] elevation.” 
Or, as Mrs. Tehillah Lichtenstein also wrote:
“…If you would know G-d, do not seek merely to prove His existence, but turn to Him with your heart; affirm your union with Him, affirm His responsiveness to prayer, pray to Him; if you actually turn to G-d … speak to Him in your heart, you will be astonished to find how close He is to you, you will feel His nearness, you will have found G-d.” 
 The Chofetz Chayim (Rabbi Yisrael Meir Kagan); Michtevai Chofetz Chayim, pp. 96-7; quoted in Pliskin, Rabbi Zelig; Gateway to Happiness, p. 255
 B’reishith/Gen. 18:12, 13
 Berachot 28b
 Leipowitz, Rabbi Yosef Zev; Nachalas Yosef, Torah; p. 125; quoted in Pliskin, Rabbi Zelig; Gateway to Happiness, p. 255
 Lichtenstein, Tehillah; Applied Judaism; “Can We Prove That G-d Exists?”; Doris Freedman, ed.; p. 96;
(originally in “How Shall We Find G-d;” Jewish Science Interpreter, June, 1940; p. 4)