While there appears to be ongoing research and debate on the nature of language in the human mind, some pertinent theories relate to the question here.
At the earliest level of linguistic learning, basic concepts about the nature of objects and relations of objects begin to form as image schemas:
In contemporary cognitive linguistics, an image schema is considered an embodied prelinguistic structure of experience that motivates conceptual metaphor mappings. Learned in early infancy they are often described as spatiotemporal relationships that enable actions and describe characteristics of the environment.
These schemas go from simple to more complex, often building upon previous schemas by combining them together into ever more composite schemas. An example low-level schema is path, or the basic concept of an object continuing in the direction it is moving. Another basic schema is contact, or the idea that one object is touching another. We can combine these two into the more complex schema of hit, or when path of one object reaches contact with another.
Moving on to language development, one particularly noteworthy set of theories is construction grammar, which basically says that all linguistic understanding stems from constructions, or abstract objects (complex schemas) with form and content. Each construction has an emergent meaning stemming from a set of simpler schemas (content) combined in a particular form. Constructions range in symbolic complexity from simple image schemas, as described above, to schemas with complex inter-scope bindings (that is, having elements of varying complexity and abstraction).
[Construction grammar] posits that there are linguistic patterns at every level of generality and specificity: from individual words, to partially filled constructions (e.g. drive X crazy), to fully abstract rules (e.g. subject–auxiliary inversion). All of these patterns are recognized as constructions. [...] The semantic meaning of a grammatical construction is made up of conceptual structures postulated in cognitive semantics: image-schemas, frames, conceptual metaphors, conceptual metonymies, prototypes of various kinds, mental spaces, and bindings across these (called "blends"). [...] Embodied construction grammar (ECG) [...] [claims] that the content of all linguistic signs involves mental simulations and is ultimately dependent on basic image schemas [...]
We can probably expect the simplest image schemas to exist in perhaps all fully developed human brains since (a) these relate concretely to basic physical phenomena and (b) most if not all human languages should have terms for common natural relationships and events. However, as the schemas in question become ever more complex, abstract, and specialised, we are bound to find discrepancies between languages. This idea is essentially the basis for linguistic relativity (also called Sapir-Whorf hypothesis), which more or less states that a person's native language either determines (strong version) or influences (weak version) his or her thinking patterns.
Research on weaker forms has produced positive empirical evidence for a relationship.
Because of these differences in higher schemas, a conversation between persons of different languages would presumably require at least some translation in meaning between comparable higher schemas. For lower schemas (such as pre-linguistic image schemas), perhaps tongue could be skipped altogether. Then again, a sufficiently advanced system may be able to transfer images or spatial models, which could serve as a work-around for certain discrepancies between languages.
A couple of important questions arise here. One, can the mental schemas or meanings behind language be activated directly, such as through a particular pattern of neural excitation? Two, upon activation of a schema or specific meaning, will the associated words or expressions be activated concurrently?
I am not prepared to answer the first question, but I have come across a related phenomenon for the second question called ideaesthesia:
[Ideaesthesia] is a neuropsychological phenomenon in which activations of concepts (inducers) evoke perception-like sensory experiences (concurrents). [...] [E]mpirical evidence indicated that most phenomena linked to synesthesia are in fact induced by semantic representations. That is, the linguistic meaning of the stimulus is what is important rather than its sensory properties.
Without evidence or good reason to the contrary, we should consider the possibility that activation of linguistic schemas or meanings may be subject to ideaesthetic effects.